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Publications mathématiques de Besançon A LGÈBRE ET THÉORIE DES NOMBRES 2011 Presses universitaires de Franche-Comté

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Prix : 40 eurosISSN en cours

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Presses universitaires de Franche-Comtéhttp://presses-ufc.univ-fcomte.fr

Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s

2011

Publ

icat

ions

mat

hém

atiq

ues

de B

esan

çon

20

11S. Ballet et R. Rolland Families of curves over finite fields

P. Bruin Computing coefficients of modular forms

B. Conrey Applications of the asymptotic large sieve

P. Lebacque et A. Zykin Asymptotic methods in number theory and algebraic geometry

M. Mohyla et G. Wiese A computational study of the asymptotic behaviour of coefficient fields of modular forms

C. Ritzenhaler Optimal curves of genus 1, 2 and 3

M. Watkins Computing with Hecke Grössencharacters

Revue du Laboratoire de mathématiques de Besançon (CNRS UMR 6623)

P r e s s e s u n i v e r s i t a i r e s d e F r a n c h e - C o m t é

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Comité de rédaction Directeur de la revue : Patrick Hild, directeur du laboratoire

Éditeur en chef : Christian Maire

Secrétaire : Catherine Pagani

Comité scientifique Bruno Anglès, Université de Caen ________________________________________________ [email protected]

Éva Bayer, École Polytechnique Fédérale de Lausanne (Suisse) ____________________________ [email protected]

Jean-Robert Belliard, Université de Franche-Comté ________________________ [email protected]

Jean-Marc Couveignes, Université Toulouse 2-Le Mirail ____________ [email protected]

Vincent Fleckinger, Université de Franche-Comté _____________________________ [email protected]

Farshid Hajir, University of Massachusetts, Amherst (USA) ____________________________ [email protected]

Nicolas Jacon, Université de Franche-Comté _______________________________________ [email protected]

Jean-François Jaulent, Université Bordeaux 1 _________________________ [email protected]

Henri Lombardi, Université de Franche-Comté __________________________________ [email protected]

Christian Maire, Université de Franche-Comté __________________________________ [email protected]

Ariane Mézard, Université de Versailles Saint-Quentin __________________________________ [email protected]

Thong Nguyen Quang Do, Université de Franche-Comté ___________________________ [email protected]

Hassan Oukhaba, Université de Franche-Comté ________________________________ [email protected]

Manabu Ozaki, Waseda University (Japon) ___________________________________________________ [email protected]

Emmanuel Royer, Université Blaise-Pascal Clermont-Ferrand 2 _______ [email protected]

Publications mathématiques de BesançonLaboratoire de Mathématiques de Besançon - UFR Sciences et Techniques - 16, route de Gray - F-25030 Besançon Cedex

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Les Publications mathématiques de Besançon éditent des articles de recherche mais aussi des articles de synthèse, des actes, des cours avancés. Les travaux soumis pour publication sont à adresser à Christian Maire [email protected] ou à l’un des membres du comité scientifique. Après acceptation, l’article devra être envoyé dans le format LaTeX 2e, de préférence avec la classe smfart. La version finale du manuscrit doit comprendre un résumé en français et un résumé en anglais.

Laboratoire de Mathématiques de Besançon (CNRS UMR 6623)

Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s - F o n d A t e u r : g e o r g e s g r A s

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Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s

2011

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© Presses universitaires de Franche-Comté, Université de Franche-Comté, 2011

Directeur de la revue : Patrick Hild, directeur du laboratoire

Éditeur en chef : Christian Maire

Secrétaire : Catherine Pagani

Laboratoire de mathématiques de Besançon (CNRS UMR 6623)

http://lmb.univ-fcomte.fr

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P r e s s e s u n i v e r s i t a i r e s d e F r a n c h e - C o m t é

Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s

Actes de lA conférence “théorie des nombres et ApplicAtions”cirm, luminy, 30 novembre-4 décembre 2009

2011

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SommaireS. Ballet et R. RollandFamilies of urves over nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18P. BruinComputing oe ients of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-36B. ConreyAppli ations of the asymptoti large sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37-46P. Leba que et A. ZykinAsymptoti methods in number theory and algebrai geometry . . . . . . . . . . . . . . . 47-73M. Mohyla et G. WieseA omputational study of the asymptoti behaviour of oe ient elds ofmodular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75-98C. RitzenthalerOptimal urves of genus 1, 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99-117M. WatkinsComputing with He ke Grössen hara ters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-135

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FAMILIES OF CURVES OVER ANY FINITE FIELD ATTAININGTHE GENERALIZED DRINFELD-VLADUT BOUNDbyStéphane Ballet & Robert RollandAbstra t. We prove the existen e of asymptoti ally exa t sequen es of algebrai fun tionelds dened over any nite eld Fq having an asymptoti ally maximum number of pla es of adegree r where r is an integer ≥ 1. It turns out that these parti ular families have an asymptoti lass number widely greater than all the La haud - Martin-Des hamps bounds when r > 1. Weemphasize that we exhibit expli it asymptoti ally exa t sequen es of algebrai fun tion eldsover any nite eld Fq, in parti ular when q is not a square, with r = 2. We expli itly onstru tan example for q = 2 and r = 4. We dedu e of this study that for any nite eld Fq there existsa sequen e of algebrai fun tion elds dened over any nite eld Fq rea hing the generalizedDrinfeld - Vladut bound.Résumé. Nous prouvons l'existen e de familles asymptotiquement exa tes de orps defon tions algébriques dénis sur un orps ni Fq qui ont un nombre maximum de pla es dedegré r où r est un entier ≥ 1. Il s'avère que pour es familles parti ulières le nombre de lasses est asymptotiquement beau oup plus grand que la borne générale de La haud - Martin-Des hamps quand r > 1. Pour r = 2 nous onstruisons expli itement des suites de orps defon tions algébriques sur tout orps ni Fq qui sont asymptotiquement exa tes et e i mêmelorsque q n'est pas un arré. Nous onstruisons un exemple pour r = 4 dans le as où q = 2.Nous déduisons de ette étude que pour tout orps ni Fq il existe une suite de orps de fon tionsalgébriques sur Fq qui atteint la borne de Drinfeld - Vladut généralisée.

1. Introdu tion1.1. General ontext and main result. When, for a given nite ground eld, thesequen e of the genus of a sequen e of algebrai fun tion elds tends to innity, there existasymptoti formulae for dierent numeri al invariants. In [10, Tsfasman generalizes someresults on the number of rational points on the urves (due to Drinfeld-Vladut [13, Ihara [5,and Serre [8) and on its Ja obian (due to Vladut [12, Rosembloom and Tsfasman [7). Hegives a formula for the asymptoti number of divisors, and some estimates for the number ofpoints in the Poin aré ltration.Key words and phrases. Finite eld, fun tion eld, asymptoti ally exa t sequen e, lass number, towerof fun tion elds.

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6 Families of urves over nite eldsFor this purpose, he introdu ed the notion of asymptoti ally exa t family of urves denedover a nite eld. It turns out that this notion is very fruitful. Indeed, for su h sequen esof urves, we an evaluate enough pre isely the asymptoti behavior of ertain numeri alinvariants, in parti ular the asymptoti number of ee tive divisors and the asymptoti lassnumber. Moreover, when these sequen es are maximal ( f. the basi inequality (1)) they havean asymptoti ally large lass number. In parti ular, it is the ase when they have a maximalnumber of pla es of a given degree. Unfortunately the existen e of su h sequen es is not knownfor any nite eld Fq, in parti ular when q is not a square ( f. Remark 5.2 in [11). In thispaper, we mainly expli itly onstru t examples of asymptoti ally exa t sequen es of algebrai fun tion elds for any nite eld, so proving the existen e of maximal asymptoti ally exa tsequen es of urves dened over any nite eld Fq where q is not a square. So, we answer inCorollary 3.4 the question asked by Tsfasman and Vladut in [11, Remark 5.2.1.2. Notation and detailed questions. Let us re all the notion of asymptoti allyexa t family of urves dened over a nite eld in the language of algebrai fun tion elds.Denition 1.1 (Asymptoti ally Exa t Sequen e). Let F/Fq = (Fk/Fq)k≥1 be a se-quen e of algebrai fun tion elds Fk/Fq dened over a nite eld Fq of genus gk = g(Fk/Fq).We suppose that the sequen e of the genus gk is an in reasing sequen e growing to innity.The sequen e F/Fq is said to be asymptoti ally exa t if for all m ≥ 1 the following limitexists:βm(F/Fq) = lim

k→+∞

Bm(Fk/Fq)

gkwhere Bm(Fk/Fq) is the number of pla es of degree m on Fk/Fq.The sequen e β = (β1, β2, ...., βm, ...) is alled the type of the asymptoti ally exa t sequen eF/Fq.Denition 1.2 (Generalized Drinfeld-Vladut Bound). Let F/Fq = (Fk/Fq)k≥1 bea asymptoti ally exa t sequen e of algebrai fun tion elds Fk/Fq dened over a nite eldFq of genus gk = g(Fk/Fq) and of type β. The sequen e β (respe tively the sequen e F/Fq)is said maximal (of maximal type) when the following basi inequality, alled GeneralizedDrinfeld-Vladut bound,(1) ∞

m=1

mβm

qm/2 − 1≤ 1is rea hed.Denition 1.3 (Drinfeld-Vladut Bound of order r). Let

Br(q, g) = maxBr(F/Fq) | F/Fq is a fun tion eld over Fq of genus g.Let us setAr(q) = lim sup

g→+∞

Br(q, g)

g.Publi ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 7We have the Drinfeld-Vladut Bound of order r :Ar(q) ≤

1

r(q

r2 − 1).Remark 1.4. Note that if for a family F/Fq of algebrai fun tion elds, there exists aninteger r su h that the Drinfeld-Vladut Bound of order r is rea hed, then this family is amaximal asymptoti ally exa t sequen e namely attaining the Generalized Drinfeld-Vladutbound. Moreover, its type is

β = (0, 0, ...., 0, βr =1

r(q

r2 − 1), 0, ...).Tsfasman and Vladut in [11 made use of these notions to obtain new general results on theasymptoti properties of zeta fun tions of urves.Note that a simple diagonal argument proves that ea h sequen e of algebrai fun tion eldsof growing genus, dened over a nite eld admits an asymptoti ally exa t subsequen e.However until now, we do not know if there exists an asymptoti ally exa t sequen e F/Fqwith a maximal β sequen e when q is not a square. Moreover, the diagonal extra tion methodis not really suitable for the two following reasons. First, in general we do not obtain by thispro ess an expli it asymptoti ally exa t sequen e of algebrai fun tion elds dened over anarbitrary nite eld, in parti ular when q is not a square. Moreover, we have no ontrol onthe growing of the genus in the extra ted sequen e of algebrai fun tion elds dened over anarbitrary nite eld. More pre isely, let us dene the notion of density of a family of algebrai fun tion elds dened over a nite eld of growing genus:Denition 1.5. Let F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds Fk/Fqof genus gk = g(Fk/Fq), dened over Fq. We suppose that the sequen e of genus gk is anin reasing sequen e growing to innity. Then, the density of the sequen e F/Fq is

d(F/Fq) = lim infk→+∞

gk

gk+1.A high density an be a useful property in some appli ations of sequen es or towers of fun tionelds. Until now, no expli it examples of dense asymptoti ally exa t sequen es F/Fq havebeen pointed out unless for the ase q square and type β = (

√q − 1, 0, · · · ).1.3. New results. In this paper, we answer the questions mentioned above. Morepre isely, for any prime power q we ontru t examples of asymptoti ally exa t sequen esattaining the Drinfeld-Vladut Bound of order 2. We dedu e that for any prime power q (inparti ular when q is not a square), there exists a maximal asymptoti ally exa t sequen esof algebrai fun tions elds namely attaining the Generalized Drinfeld-Vladut bound. Wealso onstru t for q = 2 and r = 4 an example of maximal asymptoti ally exa t sequen eattaining the Drinfeld-Vladut Bound of order 4. Then, we show that the asymptoti ally exa tsequen es of onsidered maximal type have an asymptoti ally large lass number with respe tto the La haud - Martin-Des hamps bounds. The paper is organized as follows. In se tion 2,we study the general families of asymptoti ally exa t sequen es of algebrai fun tion eldsPubli ations mathématiques de Besançon - 2011

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8 Families of urves over nite eldsdened over an arbitrary nite eld Fq of maximal type (0, ..., 1r (q

r2 − 1), 0, ..., 0, ...) where

r is an integer ≥ 1, under the assumption of their existen e. In parti ular, we study forthese general families the behavior of the asymptoti lass number hk, and we ompare ourestimation to the general known bounds of La haud - Martin-Des hamps ( f. [6). Morepre isely, we show that for su h families, if they exist, the asymptoti lass number is widelygreater than the general bounds of La haud - Martin-Des hamps when r > 1. Next, in se tion3, we onstru tively prove the existen e of asymptoti ally exa t sequen es of algebrai fun tionelds dened over any arbitrary nite eld Fq of maximal type β = (0, ..., 1r (q

r2 −1), 0, ..., 0, ...)with r = 2. Moreover, we exhibit the example of an asymptoti ally exa t sequen e of algebrai fun tion elds dened over Fq of maximal type β = (0, ..., 1

r (qr2 −1), 0, ..., 0, ...) with q = 2 and

r = 4. For this purpose, we use towers of algebrai fun tion elds oming from the des entof the densied towers of Gar ia-Sti htenoth ( f. [4 and [1). Note that all these examplesare expli it and onsist on very dense asymptoti ally exa t towers algebrai fun tion elds(maximally dense tower for q = 2 and r = 4).2. General results2.1. Parti ular families of asymptoti ally exa t sequen es. First, let us re all ertain asymptoti results. Let us rst give the following result obtained by Tsfasman in [10:Proposition 2.1. Let F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds ofin reasing genus gk growing to innity. Let f be a fun tion from N to N su h that f(gk) =

o(log(gk)). Then(2) lim supgk→+∞

1

gk

f(gk)∑

m=1

mBm(Fk)

qm/2 − 1≤ 1.Now, we an easily obtain the following result as immediate onsequen e of Proposition 2.1:Theorem 2.2. Let r be an integer ≥ 1 and F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds of in reasing genus dened over Fq su h that βr(F/Fq) = 1

r (qr2 − 1). Then

βm(F/Fq) = 0 for any integer m 6= r. In parti ular, the sequen e F/Fq is asymptoti allyexa t.Proof. Let us x m 6= r and let us prove that βm(F/Fq) = 0. We use Proposition 2.1 withthe onstant fun tion f(g) = s = max(m, r). Then we getlim supk→+∞

1

gk

s∑

j=1

jBj(Fk)

qj − 1≤ 1.But by hypothesis

lim supk→+∞

rBr(Fk)

gk(qr2 − 1)

= limk→+∞

rBr(Fk)

gk(qr2 − 1)

= 1.Publi ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 9Thenlim supgk→+∞

1

gk

1≤j≤s;j 6=r

jBj(Fk)

qj − 1= lim

gk→+∞

1

gk

1≤j≤s;j 6=r

jBj(Fk)

qj − 1= 0.But

Bm(Fk)

gk≤ (qm − 1)

m

1

gk

1≤j≤s;j 6=r

jBj(Fk)

qj − 1

.Hen elim

gk→+∞

Bm(Fk)

gk= 0.Note that for any k the following holds:

B1(Fk/Fqr) =∑

i|r

iBi(Fk/Fq).Then if βr(F/Fq) = 1r (q

r2 − 1), by Theorem 2.2 we on lude that β1(F/Fqr ) exists and that

β1(F/Fqr) = (qr2 − 1).In parti ular the sequen e F/Fqr rea hes the lassi al Drinfeld-Vladut bound and onse-quently qr is a square.If β1(F/Fqr ) exists then it does not ne essarily imply that βr(F/Fq) exists but only that

limk→+∞

P

m|r mBm(Fk/Fq)

gkexists. In fa t, this onverse depends on the dening equations ofthe algebrai fun tion elds Fk/Fq.Now, let us give a simple onsequen e of Theorem 2.2.Proposition 2.3. Let r and i be integers ≥ 1 su h that i divides r. Suppose that

F/Fq = (Fk/Fq)k≥1is an asymptoti ally exa t sequen e of algebrai fun tion elds dened over Fq of type (β1 =

0, . . . , βr−1 = 0, βr = 1r (q

r2 − 1), βr+1 = 0, . . .). Then the sequen e F/Fqi = (Fk/Fqi)k≥1 ofalgebrai fun tion eld dened over Fqi is asymptoti ally exa t of type (β1 = 0, .., β r

i−1 =

0, β ri

= ir (q

r2 − 1), β r

i+1 = 0, . . .).Proof. Let us remark that by [9, Lemma V.1.9, p. 163, if P is a pla e of degree r′ of

F/Fq, there are gcd((r′, i)) pla es of degree r′

gcd(r′,i) over P in the extension F/Fqi . As we areinterested by the pla es of degree r/i in F/Fqi , let us introdu e the setS = r′; r gcd(r′, i) = i r′ = r′; lcm (r′, i) = r.Then,

Br/i(F/Fqi) =∑

r′∈S

ir′

rBr′(F/Fq).Publi ations mathématiques de Besançon - 2011

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10 Families of urves over nite eldsWe know that all the βj(F/Fq) = 0 but βr(F/Fq) = 1r (q

r2 − 1). Then

βr/i(F/Fqi) = iβr(F/Fq),

βr/i(F/Fqi) =i

r

(

qr2 − 1

)

.2.2. Number of points of the Ja obian. Now, we are interested by the Ja obian ardinality of the asymptoti ally exa t sequen es F/Fq = (Fk/Fq)k≥1 of type (0, .., 0, 1r (q

r2 −

1), 0, ..., 0).Let us denote by hk = hk(Fk/Fq) the lass number of the algebrai fun tion eld Fk/Fq. Letus onsider the following quantities introdu ed by Tsfasman in [10:Hinf (F/Fq) = lim inf

k→+∞

1

gklog hk

Hsup(F/Fq) = lim supk→+∞

1

gklog hk.If they oin ides, we just write:

H(F/Fq) = limk→+∞

1

gklog hk = Hinf (F/Fq) = Hsup(F/Fq).Then under the assumptions of the previous se tion, we obtain the following result on thesequen e of lass numbers of these families of algebrai fun tion elds:Theorem 2.4. Let F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds of in- reasing genus dened over Fq su h that βr(F/Fq) = 1

r (qr2 − 1) where r is an integer. Then,the limit H(F/Fq) exists and we have:

H(F/Fq) = limk→+∞

1

gklog hk = log

qqr2

(qr − 1)1r(q

r2 −1)

.Proof. By Corollary 1 in [10, we know that for any asymptoti ally exa t family of algebrai fun tion elds dened over Fq, the limit H(F/Fq) exists andH(F/Fq) = lim

k→+∞

1

gklog hk = log q +

∞∑

m=1

βm. logqm

qm − 1.Hen e, the result follows from Theorem 2.2.Corollary 2.5. Let F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds of in- reasing genus dened over Fq su h that βr(F/Fq) = 1

r (qr2 − 1) where r is an integer. Thenthere exists an integer k0 su h that for any integer k ≥ k0,

hk > qgk .Publi ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 11Proof. By Theorem 2.4, we have limk→+∞(hk)1

gk = qqr2

(qr−1)1r (q

r2 −1)

. Butqq

r2

(qr − 1)1r(q

r2 −1)

>qq

r2

(qr)1r(q

r2 −1)

= q.Hen e, for a su iently large k0, we have for k ≥ k0 the following inequality(hk)

1gk > q.Let us ompare this estimation of hk to the general lower bounds given by G. La haud andM. Martin-Des hamps in [6.Theorem 2.6 (La haud - Martin-Des hamps bounds). Let X be a proje tive ir-redu ible and non-singular algebrai urve dened over the nite eld Fq of genus g. Let JXbe the Ja obian of X and h the lass number h = |JX(Fq)|. Then1. h ≥ L1 = qg−1 (q−1)2

(q+1)(g+1) ,2. h ≥ L2 =(√

q − 1)2 gg−1−1

g|X(Fq)|+q−1

q−1 ,3. if g >√

q/2 and if B1(X/Fq) ≥ 1, then the following holds:h ≥ L3 = (qg − 1)

q − 1

q + g + gq.Then we an prove that for a family of algebrai fun tion elds satisfying the onditions ofCorollary 2.5, the lass numbers hk greatly ex eeds the bounds Li. More pre iselyProposition 2.7. Let F/Fq = (Fk/Fq)k≥1 be a sequen e of algebrai fun tion elds ofin reasing genus dened over Fq su h that βr(F/Fq) = 1

r (qr2 − 1) where r is an integer. Then1. for i = 1, 3

limk→+∞hk

Li= +∞,2. for i = 2 the following holds:(a) if r>1 then

limk→+∞hk

L2= +∞,(b) if r=1 then

hk

L2≥ 2.5.Proof. 1. ase i = 1: the following holds

L1 = qgk−1 (q − 1)2

(q + 1)(gk + 1)= qgk

(q − 1)2

q(q + 1)(gk + 1)<

qgk

(gk + 1),so, using the previous orollary 2.5, we on lude that for k large

hk

L1> gk Publi ations mathématiques de Besançon - 2011

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12 Families of urves over nite eldsand onsequentlylim

k→+∞

hk

L1= +∞;2. ase i = 2:(a) ase r = 1: we bound the number of rational points using the Weil bound. Morepre isely

L2 = (q + 1 − 2√

q)qgk−1 − 1

gk

B1(Fk/Fq) + q − 1

q − 1≤

(q + 1 − 2√

q)qgk−1 − 1

gk

2q + 2gk√

q

q − 1<

2q + 1 − 2

√q

(q − 1)√

qqgk

(

1 +

√q

gk

)

;but for all q ≥ 2

2q + 1 − 2

√q

(q − 1)√

q< 0.4,then

L2 < 0.4

(

1 +

√q

gk

)

qgk ,hen ehk

L2> 2.5

gk

gk +√

qwhi h gives the result;(b) ase r > 1: in this ase we know thatlim

k→+∞

B1(Fk/Fq)

gk= β1(F/Fq) = 0.But

L2 <q + 1 − 2

√q

(q − 1)qqgk

B1(Fk/Fq) + q − 1

gk.Then

hk

L2>

(q − 1)q

q + 1 − 2√

q

gk

B1(Fk/Fq) + q − 1.We know that

limk→+∞

gk

B1(Fk/Fq) + q − 1= +∞,then

limk→+∞

hk

L2= +∞.Publi ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 133. ase i = 3:L3 = (qg − 1)

q − 1

q + g + gq<

qgk

gk,then for k large

hk

L3> gkand onsequently

limk→+∞

hk

L1= +∞.As we see, if F/Fq = (Fk/Fq)k≥1 satises the assumptions of Theorem 2.4, we have hk >

qgk > qgk−1 (q−1)2

(q+1)gkfor k ≥ k0 su iently large. In fa t, the value k0 depends at least on thevalues of r and q and we an not know anything about this value in the general ase.Remark: We an remark that the lass number of these families is very near the La haud- Martin-Des hamps bound L2 when r = 1 but is mu h greater than the La haud - Martin-Des hamps bound L2 when r > 1.3. Examples of asymptoti ally exa t towersLet us note Fq2 a nite eld with q = pr and r an integer.3.1. Sequen es F/Fq with β2(F/Fq) = 1

2(q − 1) = A2(q). We onsider the Gar ia-Sti htenoth's tower T0 over Fq2 onstru ted in [4. Re all that this tower is dened re ursivelyin the following way. We set F1 = Fq2(x1) the rational fun tion eld over Fq2 , and for i ≥ 1we deneFi+1 = Fi(zi+1),where zi+1 satises the equation

zqi+1 + zi+1 = xq+1

i ,withxi =

zi

xi−1for i ≥ 2.We onsider the ompleted Gar ia-Sti htenoth's tower T1/Fq2 dened over Fq2 studied in [2obtained from T0/Fq2 by adjun tion of intermediate steps. Namely we have

T1/Fq2 : F1,0 ⊂ · · · ⊂ Fi,0 ⊂ Fi,1 ⊂ · · · ⊂ Fi,s ⊂ · · · ⊂ Fi+1,0 ⊂ · · ·with s = 0, ..., r. Note that the steps Fi,0/Fq2 = Fi−1,r/Fq2 are the steps Fi/Fq2 of the Gar ia-Sti htenoth's tower T0/Fq2 and Fi,s/Fq2 (1 ≤ s ≤ r− 1) are the intermediate steps onsideredin [2.Let us denote by gk the genus of Fk/Fq2 in T0/Fq2 , by gk,s the genus of Fk,s in T1/Fq2 and byB1(Fk,s/Fq2) the number of pla es of degree one of Fk,s/Fq2 in T1/Fq2 .Publi ations mathématiques de Besançon - 2011

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14 Families of urves over nite eldsRe all that ea h extension Fk,s/Fk is Galois of degree ps with full onstant eld Fq2 . Moreover,we know by [3, Theorem 4.3 that the des ent of the denition eld of the tower T1/Fq2 fromFq2 to Fq is possible. More pre isely, there exists a tower T2/Fq dened over Fq given by asequen e:

T2/Fq : G1,0 ⊂ · · · ⊂ Gi,0 ⊂ Gi,1 ⊂ · · · ⊂ Gi,r−1 ⊂ Gi+1,0 ⊂ · · ·dened over the onstant eld Fq and related to the tower T1/Fq2 byFk,s = Fq2Gk,s for all k and s,namely Fk,s/Fq2 is the onstant eld extension of Gk,s/Fq. First, let us study the asymptoti behavior of degree one pla es of the fun tion elds of the tower T2/Fq and more pre iselythe existen e and the value of β1(T2/Fq). In order to derive a result on the tower T2/Fq webegin by the study of the terms given by the des ent of the lassi al Gar ia-Sti htenoth tower

T0/Fq2 . Next we will study the intermediate steps.Lemma 3.1. Let T0/Fq2 = (Fk/Fq2)k≥1 the Gar ia-Sti htenoth tower dened over Fq2and T ′0/Fq = (Gk/Fq)k≥1 its des ent over the denition eld Fq i.e su h that for any integerk, Fk = Fq2Gk. Then

β1(T′0/Fq) = lim

k→+∞

B1(Gk/Fq)

g(Gk/Fq)= 0.Proof. First, note that if the algebrai fun tion eld Fk/Fq2 is a onstant eld extensionof Gk/Fq, above any pla e of degree one in Gk/Fq there exists a unique pla e of degree onein Fk/Fq2 . Consequently, let us use the lassi ation given in [4, p. 221 of the pla es ofdegree one of Fk/Fq2 . Let us remark that the number of pla es of degree one whi h are notof type (A), is less or equal to 2q2 (see [4, Remark 3.4). Moreover, the genus gk of thealgebrai fun tion elds Gk/Fq and Fk/Fq2 is su h that gk ≥ qk by the Hurwitz theorem, thenwe an fo us our study on pla es of type (A). The pla es of type (A) are built re ursivelyin the following way ( f. [4, p. 220 and Proposition 1.1 (iv)). Let α ∈ Fq2 \ 0 and Pαbe the pla e of F1/Fq2 whi h is the zero of x1 − α. For any α ∈ Fq2 \ 0 the polynomialequation zq

2 + z2 = αq+1 has q distin t roots u1, · · · uq in Fq2 , and for ea h ui there is a uniquepla e P(α,i) of F2/Fq2 above Pα and this pla e P(α,i) is a zero of z2 − ui. We iterate nowthe pro ess starting from the pla es P(α,i) to obtain su essively the pla es of type (A) ofF3/Fq2, · · · , Fk/Fq2, · · · ; then, ea h pla e P of type (A) of Fk/Fq2 is a zero of zk − u whereu is itself a zero of uq + u = γ for some γ 6= 0 in Fq2 . Let us denote by Pu this pla e. Now,we want to ount the number of pla es P ′

u of degree one in Gk/Fq, that is to say the onlypla es whi h admit a unique pla e of degree one Pu in Fk/Fq2 lying over P ′u. First, note thatit is possible only if u is a solution in Fq of the equation uq + u = γ where γ is in Fq \ 0.Indeed, if u is not in Fq, there exists an automorphism σ in the Galois group Gal(Fk/Gk) ofthe degree two Galois extension Fk/Fq2 of Gk/Fq su h that σ(Pu) 6= Pu. Hen e, the uniquepla e of Gk/Fq lying under Pu is a pla e of degree 2. But uq + u = γ has one solution in Fqif p 6= 2 and no solution in Fq if p = 2. Hen e the number of pla es of degree one of Gk/FqPubli ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 15whi h are lying under a pla e of type (A) of Fk/Fq2 is equal to zero if p = 2 and equal toq − 1 if p 6= 2. We on lude that

limk→+∞

B1(Gk/Fq)

g(Gk/Fq)= 0.Let us remark that in any ase, the number of pla es of degree one of Gk/Fq is less or equalto 2q2.Now we an get a similar result for the des ent T2/Fq of the densied tower T1/Fq2 of T0/Fq2 .Lemma 3.2. The tower T2/Fq is su h that:

β1(T2/Fq) = limg(Gk,s/Fq)→+∞

B1(Gk,s/Fq)

g(Gk,s/Fq)= 0.Proof. As Gk+1 = Gk+1,0 is an extension of Gk,s we get B1(Gk,s/Fq) ≤ B1(Gk+1/Fq).Using the omputation done in the proof of Lemma 3.1 and Remark 3.4 in [4 we have

B1(Gk+1/Fq) ≤ 2q2, then we get B1(Gk,s/Fq) ≤ 2q2. By [2, Corollary 2.2 we know thatlim

l→+∞

g(Gl+1/Fq)

g(Gl/Fq)= pwhere g(Gl/Fq) and g(Gl+1/Fq) denote the genus of two onse utive algebrai fun tion eldsin T2/Fq. Then for k su iently large we get

gk,s ≥ gk,0 = gk.We on lude that β1(T2/Fq) = 0.Let us prove a proposition establishing that the tower T2/Fq is asymptoti ally exa t withgood density.Proposition 3.3. Let q = pr. For any integer k ≥ 1, for any integer s su h that s =

0, 1, ..., r, the algebrai fun tion eld Gk,s/Fq in the tower T2 has a genus g(Gk,s/Fq) = gk,swith B1(Gk,s/Fq) pla es of degree one, B2(Gk,s/Fq) pla es of degree two su h that:1. g(Gk,s/Fq) ≤ g(Gk+1/Fq)pr−s + 1 with g(Gk+1/Fq) = gk+1 ≤ qk+1 + qk.2. B1(Gk,s/Fq) + 2B2(Gk,s/Fq) ≥ (q2 − 1)qk−1ps.3. β2(T2/Fq) = limgk,s→+∞

B2(Gk,s/Fq)gk,s

= 12 (q − 1) = A2(q).4. d(T2/Fq) = liml→+∞

g(Gl/Fq)g(Gl+1/Fq) = 1

p where g(Gl/Fq) and g(Gl+1/Fq) denote the genus oftwo onse utive algebrai fun tion elds in T2/Fq.Proof. By Theorem 2.2 in [2, we have g(Fk,s/Fq2) ≤ g(Fk+1/Fq2 )

pr−s + 1 with g(Fk+1/Fq2) =

gk+1 ≤ qk+1 + qk . Then, as the algebrai fun tion eld Fk,s/Fq2 is a onstant eld extensionof Gk,s/Fq, for any integer k and s = 0, 1 or 2, the algebrai fun tion elds Fk,s/Fq2 andGk,s/Fq have the same genus. So, the inequality satised by the genus g(Fk,s/Fq2) is alsotrue for the genus g(Gk,s/Fq). Moreover, the number of pla es of degree one B1(Fk,s/Fq2)of Fk,s/Fq2 is su h that B1(Fk,s/Fq2) ≥ (q2 − 1)qk−1ps. Then, as the algebrai fun tion eldPubli ations mathématiques de Besançon - 2011

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16 Families of urves over nite eldsFk,s/Fq2 is a onstant eld extension of Gk,s/Fq of degree 2, it is lear that for any integerk and s, we have B1(Gk,s/Fq) + 2B2(Gk,s/Fq) ≥ (q2 − 1)qk−1ps. Moreover, we know thatβ1(T2/Fq) = 0 by Lemma 3.2. But B1(Gk,s/Fq) + 2B2(Gk,s/Fq) = B1(Fk,s/Fq2) and as by[4, β1(T1/Fq2) = A(q2), we have β2(T2/Fq) = 1

2 (q − 1).In parti ular, the following result holds:Corollary 3.4. For any prime power q, there exists a sequen e of algebrai fun tion eldsdened over the nite eld Fq rea hing the Generalized Drinfeld-Vladut bound.3.2. Sequen es F/F2 with β4(F/F2) = 34 = A4(2). We use the notations of the previ-ous paragraph on erning the towers T1/Fq2 and T2/Fq. Now we suppose that q = p2 and weask the following question: is the des ent of the denition eld of the tower T1/Fq2 from Fq2to Fp possible? The following result gives a positive answer for the ase p = 2.Proposition 3.5. Let p = 2. If q = p2, the des ent of the denition eld of the tower

T1/Fq2 from Fq2 to Fp is possible. More pre isely, there exists a tower T3/Fp dened over Fpgiven by a sequen e:T3/Fp = H1,0 ⊆ H1,1 ⊆ H2,0 ⊆ H2,1 ⊆ ...dened over the onstant eld Fp and related to the towers T1/Fq2 and T2/Fq by

Fk,s = Fq2Hk,s for all k and s = 0, 1 or 2,

Gk,s = FqHk,s for all k and s = 0, 1 or 2,namely Fk,s/Fq2 is the onstant eld extension of Gk,s/Fq and Hk,s/Fp and Gk,s/Fq is the onstant eld extension of Hk,s/Fp.Proof. Let x1 be a trans endent element over F2 and let us setH1 = F2(x1), G1 = F4(x1), F1 = F16(x1).We dene re ursively for k ≥ 11. zk+1 su h that z4

k+1 + zk+1 = x5k,2. tk+1 su h that t2k+1 + tk+1 = x5

k(or alternatively tk+1 = zk+1(zk+1 + 1)),3. xk = zk/xk−1 if k > 1 (x1 is yet dened),4. Hk,1 = Hk,0(tk+1) = Hk(tk+1), Hk+1,0 = Hk+1 = Hk(zk+1), Gk,1 = Gk,0(tk+1) =

Gk(tk+1), Gk+1,0 = Gk+1 = Gk(zk+1), Fk,1 = Fk,0(tk+1) = Fk(tk+1), Fk+1,0 = Fk+1 =

Fk(zk+1).By [3, the towerT1/Fq2 = (Fk,i/Fq2)k≥1,i=0,1is the densied Gar ia-Sti htenoth's tower over F16 and the two other towers T2/Fq and T3/Fpare respe tively the des ent of T1/Fq2 over F4 and over F2.Publi ations mathématiques de Besançon - 2011

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Stéphane Ballet and Robert Rolland 17Proposition 3.6. Let q = p2 = 4. For any integer k ≥ 1, for any integer s su h thats = 0, 1 or 2, the algebrai fun tion eld Hk,s/Fp in the tower T3 has a genus g(Hk,s/Fp) = gk,swith B1(Hk,s/Fp) pla es of degree one, B2(Hk,s/Fp) pla es of degree two and B4(Hk,s/Fp)pla es of degree 4 su h that:1. g(Hk,s/Fp) ≤ g(Hk+1/Fp)

pr−s + 1 with g(Hk+1/Fp) = gk+1 ≤ qk+1 + qk.2. B1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) ≥ (q2 − 1)qk−1ps.3. β4(T3/Fp) = limgk,s→+∞B4(Hk,s/Fp)

gk,s= 1

4(p2 − 1) = 34 = A4(2).4. d(T3/Fp) = liml→+∞

g(Hl/Fp)g(Hl+1/Fp) = 1

2 where g(Hl/Fp) and g(Hl+1/Fp) denote the genusof two onse utive algebrai fun tion elds in T3/Fp.Proof. By Theorem 2.2 in [2, we have g(Fk,s/Fq2) ≤ g(Fk+1/Fq2 )

pr−s + 1 with g(Fk+1/Fq2) =

gk+1 ≤ qk+1+qk . Then, as the algebrai fun tion eld Fk,s/Fq2 is a onstant eld extension ofHk,s/Fp, for any integer k and s = 0, 1 or 2, the algebrai fun tion elds Fk,s/Fq2 and Hk,s/Fphave the same genus. So, the inequality satised by the genus g(Fk,s/Fq2) is also true for thegenus g(Hk,s/Fp). Moreover, the number of pla es of degree one B1(Fk,s/Fq2) of Fk,s/Fq2 issu h that B1(Fk,s/Fq2) ≥ (q2 − 1)qk−1ps. Then, as the algebrai fun tion eld Fk,s/Fq2 is a onstant eld extension of Hk,s/Fp of degree 4, it is lear that for any integer k and s, we haveB1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) ≥ (q2 − 1)qk−1ps. Moreover, we have shown inthe proof of Lemma 3.2 that for any integer k ≥ 1 and any 0 ≤ s ≤ 2 the number of pla esof degree one B1(Gk,s/Fq) of Gk,s/Fq is less or equal to 2q2 and so β1(T2/Fq) = 0. Then, asthe algebrai fun tion eld Gk,s/Fq is a onstant eld extension of Hk,s/Fp of degree 2, it is lear that for any integer k and s, we have B1(Hk,s/Fp) + 2B2(Hk,s/Fp) = B1(Gk,s/Fq) andso β1(T3/Fp) = β2(T3/Fp) = 0. Moreover, B1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) =

B1(Fk,s/Fq2) and as by [4, β1(T1/Fq2) = A(q2), we have β4(T3/Fp) = A(p4) = p2 − 1.Corollary 3.7. Let T3/F2 = (Hk,s/F2)k∈N,s=0,1,2 be the tower dened above. Then thetower T3/F2 is an asymptoti ally exa t sequen e of algebrai fun tion elds dened over F2with a maximal density (for a tower).Proof. It follows from (4) of Proposition 3.3.4. Open questions1. Find asymptoti ally exa t sequen es of algebrai fun tion elds dened over any niteeld Fq, of type β = (β1, β2, ...., βm, ...) with several βi > 0, attaining the generalizedDrinfeld-Vladut bound (i.e maximal or not).2. Find asymptoti ally exa t sequen es of algebrai fun tion elds dened over any niteeld Fq, attaining the Drinfeld-Vladut bound of order r for any integer r > 2 (ex ept ase q=2 and r=4 solved in this paper). Publi ations mathématiques de Besançon - 2011

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18 Families of urves over nite elds3. Find expli it asymptoti ally exa t sequen es of algebrai fun tion elds (not Artin-S hreier type) dened over any nite eld Fq having the good pre eding properties.Referen es[1 Stéphane Ballet. Quasi-optimal algorithms for multipli ation in the extensions of F16 of degree 13,14, and 15. Journal of Pure and Applied Algebra, 171:149164, 2002.[2 Stéphane Ballet. Low in reasing tower of algebrai fun tion elds and bilinear omplexity of mul-tipli ation in any extension of Fq. Finite Fields and Their Appli ations, 9:472478, 2003.[3 Stéphane Ballet, Dominique Le Brigand, and Robert Rolland. On an appli ation of the denitioneld des ent of a tower of fun tion elds. In Pro eedings of the Conferen e Arithmeti , Geometryand Coding Theory (AGCT 2005), volume 21, pages 187203. So iété Mathématique de Fran e, sér.Séminaires et Congrès, 2009.[4 Arnaldo Gar ia and Henning Sti htenoth. A tower of artin-s hreier extensions of fun tion eldsattaining the drinfeld-vladut bound. Inventiones Mathemati ae, 121:211222, 1995.[5 Yasutaka Ihara. Some remarks on the number of rational points of algebrai urves over niteelds. journal of Fa . S i. Tokyo, 28, 721-724, 1981.[6 Gilles La haud and Mireille Martin-Des hamps. Nombre de points des ja obiennes sur un orpsnis. A ta Arithmeti a, 56(4):329340, 1990.[7 Mi hael Rosenbloom and Mi hael Tsfasman. Multipli ative latti es in global elds. InventionesMathemati ae, 17:5354, 1983.[8 Jean-Pierre Serre. The number of rationnal points on urves over nite elds, 1983. Notes by E.Bayer, Prin eton Le tures.[9 Henning Sti htenoth. Algebrai Fun tion Fields and Codes. Number 314 in Le tures Notes inMathemati s. Springer-Verlag, 1993.[10 Mi hael Tsfasman. Some remarks on the asymptoti number of points. In H. Sti htenoth andM.A. Tsfasman, editors, Coding Theory and Algebrai Geometry, volume 1518 of Le ture Notes inMathemati s, pages 178192, Berlin, 1992. Springer-Verlag. Pro eedings of AGCT-3 onferen e, June17-21, 1991, Luminy.[11 Mi hael Tsfasman and Serguei Vladut. Asymptoti properties of zeta-fun tions. Journal of Math-emati al S ien es, 84(5):14451467, 1997.[12 Serguei Vladut. An exhaustion bound for algebrai -geometri modular odes. Problems of Infor-mation Transmission, 23:2234, 1987.[13 Serguei Vladut and Vladimir Drinfeld. Number of points of an algebrai urve. Funktsional Anali Prilozhen, 17:5354, 1983.17 mai 2010Stéphane Ballet, Institut de Mathématiques de Luminy, Case 907, 13288 Marseille edex 9E-mail : stephane.balletunivmed.frRobert Rolland, Institut de Mathématiques de Luminy, Case 907, 13288 Marseille edex 9 et Asso iationACrypTA • E-mail : robert.rollanda rypta.fr

Publi ations mathématiques de Besançon - 2011

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COMPUTING COEFFICIENTS OF MODULAR FORMSbyPeter BruinAbstra t. We prove that oe ients of q-expansions of modular forms an be omputedin polynomial time under ertain assumptions, the most important of whi h is the Riemannhypothesis for ζ-fun tions of number elds. We give appli ations to omputing He ke operators, ounting points on modular urves over nite elds, and omputing the number of representa-tions of an integer as a sum of a given number of squares.Résumé (Sur le al ul des oe ients des formes modulaires). On démontre queles oe ients des q-développements des formes modulaires peuvent être al ulés en tempspolynomial sous ertaines onditions, dont la plus importante est l'hypothèse de Riemann pourles fon tions ζ des orps de nombres. On donne des appli ations aux problèmes suivants : al ulerdes opérateurs de He ke ; ompter le nombre de points d'une ourbe modulaire sur un orps ni ; al uler le nombre de représentations d'un entier omme somme d'un nombre donné de arrés.1. Introdu tionLet n and k be positive integers, and let Mk(Γ1(n)) be the omplex ve tor spa e of modularforms of weight k for the group Γ1(n). A modular form f ∈ Mk(Γ1(n)) is determined by n, kand its q-expansion oe ients am(f) for 0 ≤ m ≤ k · d(Γ1(n)), where d(Γ1(n)) is a fun tiongrowing roughly quadrati ally in n.A natural question to ask is whether, given am(f) for 0 ≤ m ≤ k ·d(Γ1(n)), one an e iently ompute am(f) for large m. In the ase n = 1, Couveignes, Edixhoven et al. [2 des ribeda deterministi algorithm that a omplishes this in time polynomial in logm for xed k.Under the generalised Riemann hypothesis, their algorithm runs in time polynomial in kand logm. Earlier algorithms, based on modular symbols, require time polynomial in m. The2000 Mathemati s Subje t Classi ation. 11E25, 11F11, 11F30, 11F80, 11Y16.Key words and phrases. Algorithms, He ke algebras, modular forms, sums of squares.The results of this arti le are based on those of my thesis [1. I am mu h indebted to my advisors BasEdixhoven and Robin de Jong for their support. I would also like to thank the organisers of the onferen eThéorie des nombres et appli ations for the opportunity to speak about this subje t, whi h has led to thisarti le.The resear h for this arti le was supported by the Netherlands Organisation for S ienti Resear h.

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20 Computing oe ients of modular formsmethod of [2 is, very briey, to ompute two-dimensional Galois representations asso iatedto eigenforms of level 1 over nite elds.In this arti le, results from the author's thesis [1 on omputing Galois representations asso i-ated to eigenforms of higher levels are used to generalise the result of Couveignes, Edixhovenet al., be it that we an urrently only give a probabilisti algorithm. The pre ise resultfrom [1 that we need is Theorem 3.1 below. We will use this to prove our main result, whi hreads as follows.Theorem 1.1. Let n0 be a positive integer. There exists a probabilisti algorithm that,given a positive integer k, a squarefree positive integer n1 oprime to n0, a number eld K, a modular form f of weight k for Γ1(n) over K, where n = n0n1, and a positive integer m in fa tored form, omputes am(f), and whose expe ted running time is bounded by a polynomial in the lengthof the input under the Riemann hypothesis for ζ-fun tions of number elds.Let us make pre ise how the number eld K and the form f should be given to the algorithmand how it returns am(f). We represent K by its multipli ation table with respe t to someQ-basis (b1, . . . , br) of K. By this we mean the rational numbers ci,j,k with 1 ≤ i, j, k ≤ rsu h that

bibj =r∑

k=1

ci,j,kbk.We represent elements of K as Q-linear ombinations of (b1, . . . , br). We represent f by its oe ients a0(f), . . . , ak·d(Γ1(n))(f); these, as well as the output am(f), are elements of K.We should also make pre ise what the word `probabilisti ' in Theorem 1.1 means. The orre tinterpretation is that the result is guaranteed to be orre t, but that the running time dependson random hoi es made during exe ution. Probabilisti algorithms with this property are ommonly alled Las Vegas algorithms. These are to be ontrasted with Monte Carlo algo-rithms, where the randomness inuen es the orre tness of the output instead of the runningtime. It is worth emphasising that the expe ted running time is dened by averaging onlyover the random hoi es made during exe ution, not over the possible inputs. For any inputx, the a tual running time of the algorithm given this input an be modelled as a randomvariable Tx. The laim that the expe ted running time is polynomial in the length of theinput means that there exists a polynomial P su h that for any input x, the expe tation of Txis at most P (length of x). We refer to Lenstra and Pomeran e [10, § 12 for an enlighteningdis ussion of probabilisti algorithms.Remark 1.2. The length of the input depends not only on k, n0, n1, logm and K, butalso on the omplexity of the given oe ients of the modular form f . For example, if f is aprimitive form f0 multiplied by an integer A, then for xed f0 and A tending to ∞, the lengthPubli ations mathématiques de Besançon - 2011

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Peter Bruin 21of the input in reases approximately by a multiple of logA, and the running time in reasesapproximately by a polynomial in logA.Remark 1.3. Without the generalised Riemann hypothesis, we are only able to provethat the running time of our algorithm is polynomial in exp(n1), exp(k) and the length of theinput. In other words, we are still able to prove un onditionally that if not only n0, but alson1 and k are xed, then the expe ted running time is polynomial in the length of the input.Remark 1.4. Omitting the ondition that m be given in fa tored form would be equiv-alent to laiming that integers that are produ ts of two prime numbers an be fa tored inpolynomial time. Namely, suppose that the theorem holds without this ondition. Applyingthe hypotheti al stronger version of the theorem with k a xed even integer greater than 2, n0 = n1 = 1, K = Q, f = Ek, the lassi al Eisenstein series Ek of weight k for Γ1(1) = SL2(Z), and m = pq, where p and q are two distin t prime numbers,we on lude that there exists a probabilisti algorithm that omputes am(Ek) in time poly-nomial in logm. From the formula

am(Ek) =∑

d|m

dk−1

= 1 + pk−1 + qk−1 +mk−1,it follows that pk−1, qk−1 an be omputed qui kly as the set of roots of the polynomialx2− (am(Ek)−mk−1−1)x+mk−1 ∈ Z[x]. Hen e we would be able to ompute p, q from min time polynomial in logm, whi h is a laim we ertainly do not wish to make.Remark 1.5. The reason why our algorithm is probabilisti is that this is the urrentstate of aairs for the algorithm to whi h Theorem 3.1 refers. This algorithm an perhapsbe turned into a deterministi one by repla ing the arithmeti over nite elds that is usedin [1 by approximate arithmeti over the omplex numbers. The latter approa h is taken byCouveignes, Edixhoven et al. [2, Chapter 12 for modular forms of level 1. There are urrentlystill some di ulties with this approa h for modular forms of higher level. We refer to [1,Introdu tion for a dis ussion of these.Remark 1.6. It would be more satisfa tory if we ould prove the theorem with the levelranging over all positive integers n. We urrently annot do this for the following reason.The modular urve X1(n) has a regular and semi-stable model over the ring of integers ZLof a suitable number eld L, but in general we do not know a good bound on the number ofirredu ible omponents of the geometri bres of su h a model at primes of ZL that divide n.If we ould prove the theorem in this more general form, then the restri tion to modular formsfor ongruen e subgroups of the form Γ1(n) ould also be removed. The reason for this is thatPubli ations mathématiques de Besançon - 2011

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22 Computing oe ients of modular formsthe spa e of modular forms of weight k for the prin ipal ongruen e subgroup Γ(n) ⊆ SL2(Z) an be embedded into Mk(Γ1(n2)) by a map that on q-expansions is given by q 7→ qn.We now turn to some appli ations of Theorem 1.1. We will prove that there exist probabilis-ti algorithms that solve the following problems in expe ted polynomial time in the input,assuming the Riemann hypothesis for ζ-fun tions of number elds: Given a positive integer k, a squarefree positive integer n and a positive integer m infa tored form, ompute the matrix of the He ke operator Tm in T(Mk(Γ1(n))) withrespe t to a xed Z-basis of T(Mk(Γ1(n))). Given a squarefree positive integer n and a prime number p ∤ n, ompute the zetafun tion of the modular urve X1(n) over Fp. Given an even positive integer k and a positive integer m in fa tored form, ompute thenumber of ways in whi h m an be written as a sum of k squares of integers.A tually, we do not prove our results in exa tly the same order as presented above. Werst prove Theorem 1.1 in the spe ial ase where f is an Eisenstein series or a primitive usp form. This su es to solve (a slightly more general version of) the above problem of omputing He ke operators. We then prove Theorem 1.1 in general. Finally, we show how tosolve the problems of omputing zeta fun tions of modular urves and nding the number ofrepresentations of an integer as a sum of squares.To on lude this introdu tion, we remark that in order to keep this arti le at a reasonablelength, we have omitted, or only briey tou hed upon, mu h material that an be found in[1 and [2. This means that the ontents of this arti le are largely disjoint from those of [1and [2. 2. Ba kgroundWe begin by olle ting the ne essary preliminaries and introdu ing our notation. For deni-tions and more ba kground, we refer to the many texts on modular forms, su h as Diamondand Im [5 or Diamond and Shurman [6.2.1. Modular forms. Let n and k be positive integers. Let Mk(Γ1(n)) denote the

C-ve tor spa e of modular forms of weight k for the groupΓ1(n) =

(

a

c

b

d

)

∈ SL2(Z)

a ≡ d ≡ 1 (mod n),

c ≡ 0 (mod n)

.For every f ∈ Mk(Γ1(n)) and every m ≥ 0, we write am(f) for the oe ient of qm in theq-expansion of f , so the q-expansion of f is the power series ∑∞

m=0 am(f)qm in K[[q]]. Forevery divisor d of n and every divisor e of n/d, there exists an inje tive C-linear mapbd,ne : Mk(Γ1(d)) Mk(Γ1(n))that, on q-expansions, has the ee t of sending q to qe.Publi ations mathématiques de Besançon - 2011

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Peter Bruin 23We dene(1) d(Γ1(n)) =1

12[SL2(Z) : ±1Γ1(n)].This d(Γ1(n)) grows roughly quadrati ally in n. A basi fa t that we will need often is thefollowing.Lemma 2.1. Any f ∈ Mk(Γ1(n)) is determined by n, k and the oe ients am(f) for

0 ≤ m ≤ k · d(Γ1(n)).Proof. If n ≥ 5, then d(Γ1(n)) is the degree of the line bundle ω of modular forms ofweight 1 on the modular urve X1(n). In that ase, we an view modular forms as globalse tions of ω⊗k. If f, g ∈ Mk(Γ1(n)) are su h that am(f) = am(g) for 0 ≤ m ≤ k · d(Γ1(n)),then f−g has a zero of order at least k ·d(Γ1(n))+1 at the usp ∞ of X1(n), and we on ludethat f = g. One an prove the lemma in general by redu ing to the ase n ≥ 5. We refer toSturm [14 for a full proof.2.2. He ke algebras. Let T(Mk(Γ1(n))) be the He ke algebra on Mk(Γ1(n)). This isa ommutative ring, free of nite rank as a Z-module and generated as a Z-algebra by theHe ke operators Tm for m ∈ 1, 2, . . . and the diamond operators 〈d〉 for d ∈ (Z/nZ)×. Ita ts on the C-ve tor spa e Mk(Γ1(n)) of modular forms.Let us give some useful formulae. We have(2) Tm1m2 = Tm1Tm2 if gcd(m1,m2) = 1and(3) Tpi+2 = TpTpi+1 − pk−1〈p〉Tpi (p prime and i ≥ 0),where 〈p〉 is to be interpreted as 0 if p divides n. For all f ∈ Mk(Γ1(n)), we have(4) am(Tp(f)) = apm(f) + pk−1am/p(〈p〉f) (p prime and m ≥ 1),where the se ond term is 0 if p divides n or if p does not divide m, and(5) a1(Tmf) = am(f) (m ≥ 1).There exists a anoni al bilinear mapT(Mk(Γ1(n))) × Mk(Γ1(n)) −→ C

(t, f) 7−→ a1(tf),indu ing an isomorphism(6) Mk(Γ1(n))∼−→ HomZ-modules(T(Mk(Γ1(n))),C)of C ⊗Z T(Mk(Γ1(n)))-modules.An eigenform of weight k for Γ1(n) is an element of Mk(Γ1(n)) spanning a one-dimensionaleigenspa e for the a tion of T(Mk(Γ1(n))). Let f be su h a form. Then a1(f) 6= 0, and wemay s ale f su h that a1(f) = 1. Now (5) implies that

Tmf = am(f)f for all m ≥ 1.Publi ations mathématiques de Besançon - 2011

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24 Computing oe ients of modular formsFurthermore, there exists a unique group homomorphismǫ : (Z/nZ)× → C×, alled the hara ter of f , su h that

〈d〉f = ǫ(d)f for all d ∈ (Z/nZ)×.Under the isomorphism (6), the eigenforms f ∈ Mk(Γ1(n)) with a1(f) = 1 orrespond to thering homomorphisms T(Mk(Γ1(n))) → C.The C-ve tor spa e Mk(Γ1(n)) an be written as a dire t sumMk(Γ1(n)) = Ek(Γ1(n)) ⊕ Sk(Γ1(n)).Here Sk(Γ1(n)) denotes the subspa e of usp forms and Ek(Γ1(n)) denotes the subspa eof Eisenstein series. The a tion of T(Mk(Γ1(n))) respe ts these subspa es, and we get a orresponding de omposition

T(Mk(Γ1(n))) = T(Ek(Γ1(n))) × T(Sk(Γ1(n)))of Z-algebras.2.3. Eisenstein series. Let d1 and d2 be positive integers su h that d1d2 divides n, and onsider primitive hara tersǫ1 : (Z/d1Z)× → C×, ǫ2 : (Z/d2Z)× → C×.(A hara ter ǫ : (Z/dZ)× → C×, with d a positive integer, is alled primitive if there is nostri t divisor e | d su h that ǫ fa tors through the quotient (Z/dZ)× → (Z/eZ)×.) We denethe formal power series(7) Eǫ1,ǫ2

k (q) = −δd1,1Bǫ2

k

2k+

∞∑

m=1

(

d|m

ǫ1(m/d)ǫ2(d)dk−1

)

qm ∈ C[[q]].Here Bǫ2k is a generalised Bernoulli number and δd1,1 is 1 or 0 depending on whether d1 = 1or d1 > 1.If k 6= 2, or if k = 2 and at least one of ǫ1 and ǫ2 is non-trivial, then Eǫ1,ǫ2

k (q) is the q-expansion of an eigenform in Ek(Γ1(d1d2)) with hara ter ǫ1ǫ2. For any divisor e of n/(d1d2),the map bd1d2,ne : Mk(Γ1(d1d2)) → Mk(Γ1(n)) sends this form to an element of Mk(Γ1(n)) with

q-expansion Eǫ1,ǫ2k (qe). As for the ase where k = 2 and both ǫ1 and ǫ2 are trivial, for everydivisor e | n with e > 1 there is an element of E2(Γ1(n)) with q-expansion E2(q) − eE2(q

e),where E2(q) is the power series(8) E2(q) = − 1

24+

∞∑

m=1

(

d|m

d

)

qm.For k 6= 2, the nite set(9) Fk(Γ1(n)) =⊔

d1d2|n

e|(n/d1d2)

Eǫ1,ǫ2k (qe) | ǫi : (Z/diZ)× → C× primitivePubli ations mathématiques de Besançon - 2011

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Peter Bruin 25is a C-basis of Ek(Γ1(n)). For k = 2, we take all Eǫ1ǫ2k (qe) for ǫ1, ǫ2 not both trivial, togetherwith the E2(q) − eE2(q

e) for all e | n with e > 1.2.4. Cusp forms. We write Snewk (Γ1(n)) for the orthogonal omplement, with respe tto the Petersson inner produ t, of the subspa e of Sk(Γ1(n)) spanned by the images ofall the bd,n

e with d stri tly dividing n. The spa e Snewk (Γ1(n)) is preserved by the a tionof T(Sk(Γ1(n))). The unique quotient of T(Sk(Γ1(n))) that a ts faithfully on Snew

k (Γ1(n)) isdenoted by T(Snewk (Γ1(n))).An eigenform f ∈ Snew

k (Γ1(n)) with a1(f) = 1 is alled a primitive usp form. The nite set(10) Bk(Γ1(n)) =⊔

d|n

e|(n/d)

bd,ne

primitive usp forms in Snewk (Γ1(n))

is a C-basis for Sk(Γ1(n)).2.5. Modular forms over other rings. We deneMint

k (Γ1(n)) = forms in Mk(Γ1(n)) with q-expansion in Z[[q]].This is a T(Mk(Γ1(n)))-module that is free of nite rank as a Z-module. For any ommutativeZ[1/n]-algebra R, we dene the R-module of modular forms of weight k for Γ1(n) with oe ients in R as

Mk(Γ1(n), R) = R⊗Z Mintk (Γ1(n)).Apart from the omplex numbers, the important examples for us are number elds and niteelds of hara teristi not dividing n. If R is any eld of hara teristi not dividing n, wedene eigenforms over R in the same way as in the ase R = C.If R is a sub-Z[1/n]-algebra of C, we identify Mk(Γ1(n), R) with the submodule of Mk(Γ1(n)) onsisting of forms with q-expansion in R[[q]].3. Modular Galois representationsLet n and k be positive integers, let F be a nite eld of hara teristi not dividing n, andlet f ∈ Mk(Γ1(n),F) be an eigenform over F.It follows from work of Ei hler, Shimura, Igusa, Deligne and Serre that there exists a ontin-uous semi-simple representation

ρf : Gal(Q/Q) → AutF Vf ,where Vf is a two-dimensional F-ve tor spa e, with the following properties: ρf is unramied at all prime numbers p not dividing nl; if p is su h a prime number, then the hara teristi polynomial of the Frobenius onju-ga y lass at p equals t2 − ap(f)t+ ǫ(p)pk−1, where ǫ is the hara ter of f .Publi ations mathématiques de Besançon - 2011

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26 Computing oe ients of modular formsThis ρf is unique up to isomorphism.The end produ t of [1 is a probabilisti algorithm for omputing representations of theform ρf , where f is an eigenform over a nite eld F. This allows us to state the followingtheorem.Theorem 3.1. Let n0 be a positive integer. There exists a probabilisti algorithm that,given a positive integer k, a squarefree positive integer n1 oprime to n0, a nite eld F of hara teristi greater than k, and an eigenform f ∈ Mk(Γ1(n)), given by its oe ients am(f) for 0 ≤ m ≤ k · d(Γ1(n)), omputes ρf in the form of the following data: the nite Galois extension Kf of Q su h that ρf fa tors asGal(Q/Q) ։ Gal(Kf/Q) AutF Vf ,given by the multipli ation table of some Q-basis (b1, . . . , br) of Kf ; for every σ ∈ Gal(Kf/Q), the matrix of σ with respe t to the basis (b1, . . . , br) and thematrix of ρf (σ) with respe t to some xed F-basis of Vf ,and that runs in expe ted time polynomial in k, n1 and #F.Moreover, on e ρf has been omputed, one an ompute ρf (Frobp) using a deterministi al-gorithm in time polynomial in k, n1, #F and log p.Remark 3.2. This running time is optimal from a ertain perspe tive, given the fa t thatthe length of the input and output of su h an algorithm is ne essarily at least polynomial in

k, n1 and #F (and log p for the se ond part).4. Some boundsIn this se tion we olle t some bounds that we will need in § 5 below to prove Theorem 1.1.4.1. The dis riminant of the new quotient of the He ke algebra. Let n and kbe positive integers. The Z-algebra T(Snewk (Γ1(n))) is redu ed, be ause there is a basis ofeigenforms for its a tion on Snew

k (Γ1(n)). Furthermore, it is free of nite rank as a Z-module.In parti ular, it has a non-zero dis riminant discT(Snewk (Γ1(n))).Lemma 4.1. The logarithm of |discT(Snew

k (Γ1(n)))| is bounded by a polynomial in nand k.Proof. The method of Ullmo [15, who onsidered usp forms of weight 2 for Γ0(n) with nsquarefree, extends without di ulty to our situation. For ompleteness, let us give a proofin this more general setting.We abbreviateT = T(Snew

k (Γ1(n)))Publi ations mathématiques de Besançon - 2011

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Peter Bruin 27andr = dimC Snew

k (Γ1(n))

= rankZ T.It follows from Lemma 2.1 and (6) that the Q-ve tor spa e Q⊗ZT is spanned by the elementsT1, . . . , Tk·d(Γ1(n)), with d(Γ1(n)) as in (1). We an therefore hoose integers

1 ≤ m1 ≤ · · · ≤ mr ≤ k · d(Γ1(n))su h that the elements Tm1 , . . . , Tmr of T are Z-linearly independent. We let T′ denote thesubgroup of T spanned by Tm1 , . . . , Tmr . This T′ is free of rank r as a Z-module, so it hasnite index (T : T′) in T, anddiscT =

discT′

(T′ : T)2.In parti ular, this implies

|discT| ≤∣

∣discT′∣

∣ .We next use the denition of the dis riminant:discT′ = det

(

tr(TmuTmv )ru,v=1

)

,where tr(e) denotes the tra e of the Z-linear map T′ → T′ sending t to et. Now the tra eof an endomorphism e of T′ equals the tra e of the endomorphism dual to e on the C-ve torspa eHomZ-modules(T

′,C) ∼= Snewk (Γ1(n)).We let f1, . . . , fr be the primitive usp forms in Snew

k (Γ1(n)), and we abbreviateαt,u = amt(fu).Then we get

tr(TmuTmv ) =r∑

t=1

αt,uαt,v .We then ompute discT′ as follows:discT′ = det

( r∑

t=1

αt,uαt,v

)r

u,v=1

= det

α1,1 α2,1 . . . αr,1

α1,2 α2,2 . . . αr,2... ... . . . ...α1,r α2,r . . . αr,r

α1,1 α1,2 . . . α1,r

α2,1 α2,2 . . . α2,r... ... . . . ...αr,1 αr,2 . . . αr,r

= det(

(αt,u)rt,u=1

)2. Publi ations mathématiques de Besançon - 2011

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28 Computing oe ients of modular formsDeligne's bound for the oe ients of eigenforms, proved in [3 and [4, implies the inequality|αt,u| = |amt(fu)|

≤ σ0(mt)m(k−1)/2t .Here σ0(m) denotes the number of positive divisors of m. Elementary estimates now showthat log |discT| is bounded by a polynomial in n and k.4.2. Primes of small norm in number elds. The Riemann hypothesis for the ζ-fun tion of a number eld K has the following well-known impli ation of for the existen e ofprime ideals of small norm in the ring of integers of K.Lemma 4.2. Let ǫ and δ be positive real numbers. There exist positive real numbers

A and B su h that the following holds. Let K be an number eld su h that the Riemannhypothesis is true for the ζ-fun tion of K. Let ZK denote the ring of integers of K, and forevery prime number p let λK(p) denote the number of prime ideals of ZK of norm equal to p.Then for all real numbers x ≥ 2 su h thatxδ

(log x)2≥ A[K : Q] and xδ

log x≥ B log |discZK |we have

p≤x primeλK(p) log p− x

≤ ǫx1/2+δ .Proof. For every prime number p and every positive integer m, we deneΛK(pm) =

t|m

t · #prime ideals of norm pt in ZK · log p.In parti ular, this implies ΛK(p) = λK(p) log p for every prime number p. We dene ΛK(n) =

0 if n is not a prime power. The relation between ζK and ΛK is the Diri hlet series−ζ

′K(s)

ζK(s)=

∞∑

n=1

ΛK(n)n−s.We deneψK : [1,∞) −→ R

x 7→∑

n≤x

ΛK(n).Now there exists a positive real number c, independent of K, su h that the generalised Rie-mann hypothesis for ζK implies the estimate|ψK(x) − x| ≤ c

√x log(x) log

(

x[K:Q] |discZK |) for all x ≥ 2;Publi ations mathématiques de Besançon - 2011

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Peter Bruin 29see Iwanie and Kowalski [9, Theorem 5.15. By elementary arguments, it follows that thereexists a positive real number c′, also independent of K, su h that∣

p≤x primeλK(p) log p− x

≤ c′√x log(x) log

(

x[K:Q] |discZK |) for all x ≥ 2.It is now straightforward to he k that taking A = B = 2c′/ǫ works.5. Proof of Theorem 1.1As already mentioned briey in the introdu tion, Theorem 1.1 will be proved as follows. Werst prove the following basi ases: f is an element of the form Eǫ1,ǫ2

k in Ek(Γ1(d1d2)), where ǫ1 : (Z/d1Z)× → C× andǫ2 : (Z/d2Z)× → C× are primitive hara ters; f is a primitive usp form in Sk(Γ1(n)).In ea h ase, we take K to be the number eld generated by the oe ients of f , and weassume that m is a prime number. After proving these spe ial ases, we show that we an ompute the He ke algebra T(Mk(Γ1(n))) in a sense that will be explained in § 5.3 below. Itis then straightforward to dedu e Theorem 1.1 in general.5.1. Eisenstein series. We start by onsidering the Eisenstein series Eǫ1,ǫ2

k , whereǫ1 : (Z/d1Z)× → C× and ǫ2 : (Z/d2Z)× → C× are primitive hara ters and e is a divisorof n/(d1d2). For onvenien e, we also allow the ase of the `pseudo-Eisenstein series' E2 de-ned by (8). LetK be the y lotomi extension of Q generated by the images of ǫ1 and ǫ2. Theformula (7) shows that for every prime number p, we an ompute the element ap(E

ǫ1,ǫ2k ) ∈ Kin time polynomial in n, k and log p.5.2. Primitive forms. We ontinue with the ase where f is a primitive usp formin Snew

k (Γ1(n)) and K is the number eld generated by the oe ients of f . Let ZK denotethe ring of integers of K. There exists a unique ring homomorphismef : T(Snew

k (Γ1(n))) → ZKsending ea h He ke operator to its eigenvalue on f . Let A denote the image of ef . It is ofnite index (ZK : A) in ZK , and we havediscA = (ZK : A)2 discZK .Furthermore, we have

|discA| ≤ |discT(Snewk (Γ1(n)))| and [K : Q] ≤ rankZ T(Snew

k (Γ1(n))).Lemma 4.1 now implies that log |discA|, and hen e also log |discZK | and log(ZK : A), arebounded by a polynomial in n and k. The same learly holds for [K : Q].Publi ations mathématiques de Besançon - 2011

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30 Computing oe ients of modular formsNow let p be a prime number. We have to show that we an ompute ap(f) in time polynomialin n, k and log p. In Couveignes, Edixhoven et al. [2, § 15.2 it is explained in detail how todo this. We only give a sket h.We may assume that p does not divide n; namely, if p does divide n, then we an spend timepolynomial in p, so using modular symbols is fast enough; see § 5.3 below.By Lemma 4.2 applied to K and the fa t that log(ZK : A) is bounded by a polynomial in nand k, we an hoose x su iently large, but bounded by a polynomial in n and k, su h thatif M is the set of maximal ideals of A whose norm is a prime number lying in the interval(k, x] and dierent from p, we have(11) ∏

m∈M

Norm(m) ≥(

2([K:Q]+1)/2 · 2p(k−1)/2)[K:Q]

.An explanation for the right-hand side will be given below. We ompute ap(f) using thefollowing algorithm.1. Compute a Z-basis for A.2. Compute a bound x and the set M of maximal ideals of A su h that the set M denedabove satises (11).3. For all m ∈ M , ompute the Galois representation ρf mod m : Gal(Q/Q) → GL2(A/m)using Theorem 3.1.4. For all m ∈M , ompute(ap(f) mod m) = tr(ρf mod m(Frobp)) ∈ A/m,again using Theorem 3.1.5. Compute an LLL-redu ed Z-basis for the ideal a =

m∈M m of A.6. From the ap(f) mod m, ompute the image of ap(f) in A/a.7. Using the LLL algorithm, re onstru t ap(f) as the shortest representative in A of theimage of ap(f) in A/a. This works be ause of the inequality (11).5.3. Computing He ke operators. We represent T(Mk(Γ1(n))) in the following form:we spe ify its multipli ation table with respe t to a suitable Z-basis (b1, . . . , br), togetherwith the He ke operators Tm for 1 ≤ m ≤ k · d(Γ1(n)) and the diamond operators 〈d〉 forall d ∈ (Z/nZ)× as Z-linear ombinations of (b1, . . . , br). These data spe ify T(Mk(Γ1(n)))uniquely be ause the above operators generate T(Mk(Γ1(n))). In other words, if the samedata are given with respe t to a dierent basis of T(Mk(Γ1(n))), there exists exa tly one hange of Z-basis ompatible with the given Tm and 〈d〉.Theorem 5.1. Let n0 be a positive integer. There exists a probabilisti algorithm that,given a positive integer k, a squarefree positive integer n1 oprime to n0, and a positive integer m in fa tored form, omputesPubli ations mathématiques de Besançon - 2011

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Peter Bruin 31 the He ke algebra T(Mk(Γ1(n))) as above, where n = n0n1, and the element Tm on the basis (b1, . . . , br),and that runs in expe ted time polynomial in k, n1 and logm under the Riemann hypothesisfor ζ-fun tions of number elds.Proof. We need some more information about the a tion of He ke operators on q-expansions. As a basis for Mk(Γ1(n)) we take the union of the basis Fk(Γ1(n)) of Ek(Γ1(n))dened by (9) and the basis Bk(Γ1(n)) of Sk(Γ1(n)) dened by (10).Let f be either an Eisenstein series Eǫ1,ǫ2k ∈ Ek(Γ1(d1d2)) as above or a primitive formin Sk(Γ1(d)). In the rst ase, we put d = d1d2. The formula (4) for the a tion of the He keoperator Tp shows that the relation between Tp and the maps bd,n

e : Mk(Γ1(d)) → Mk(Γ1(n)),where e runs through the divisors of n/d, is as follows:(12) Tp(bd,ne f) =

ap · bd,ne f if p ∤ n;

bd,ne/pf if p | e;ap · bd,n

e f − pk−1ǫ(p)bd,npe f if p ∤ d, p ∤ e and p | n;

ap · bd,ne f if p | d and p ∤ e.This formula gives the matrix of Tp with respe t to the basis Fk(Γ1(n)) of Ek(Γ1(n)) and thebasis Bk(Γ1(n)) of Sk(Γ1(n)).We rst ompute the q-expansions of the Eisenstein series Eǫ1,ǫ2

k ∈ Ek(Γ1(d1d2)), withǫi : (Z/diZ)× → C× primitive hara ters su h that d1d2 | n, as in § 2.3. From these q-expansions and (12) we then ompute the He ke algebra T(Ek(Γ1(n))) in the form des ribedabove in time polynomial in n and k.Given a prime number p, we ompute all the ap(E

ǫ1,ǫ2k ) as in § 5.1, and we nd the matrixof Tp using (12). We then express Tp on the basis of T(Ek(Γ1(n))) that we omputed earlier.In this way, we an ompute the He ke operator Tp ∈ T(Ek(Γ1(n))) in time polynomial in n,

k and log p.For usp forms, the q-expansions are omputed from the He ke algebra instead of vi e versa.We ompute the He ke algebras T(Sk(Γ1(d))), where d runs through the divisors of n, in theform des ribed above. These data an be omputed in time polynomial in n and k usingdeterministi algorithms based on modular symbols and the LLL latti e basis redu tion algo-rithm; see Stein [13, Chapter 8 and the author's thesis [1, § IV.4.1. From ea h T(Sk(Γ1(d))),we ompute the q-expansions of the primitive usp forms in Sk(Γ1(d)).So far, we have only used existing methods. To ompute the He ke operator Tp ∈ T(Sk(Γ1(n)))for a prime number p in time polynomial in log p, we need our new tools. For every divisord of n and every primitive form f ∈ Sk(Γ1(d)), we ompute ap(f) as in § 5.2. Using (12), weobtain the matrix of Tp with respe t to the basis Bk(Γ1(n)). We nally express Tp on thebasis of T(Sk(Γ1(n))) that we omputed earlier. Publi ations mathématiques de Besançon - 2011

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32 Computing oe ients of modular formsNow let m be an arbitrary positive integer, and suppose that we know the fa torisation of m.Then we an ompute the elementTm ∈ T(Mk(Γ1(n))) = T(Ek(Γ1(n))) × T(Sk(Γ1(n)))from the Tp for for p | m prime in time polynomial in logm using the identities (2) and (3).5.4. Proof of Theorem 1.1 in general. Given n, k, K, f and m as in the theorem,we ompute am(f) as follows. We rst ompute T(Mk(Γ1(n))) using modular symbols. From

f we then determine the unique Z-linear mapef : T(Mk(Γ1(n))) → Ksending Ti to ai(f) for all i with 1 ≤ i ≤ k · d(Γ1(n)). Using Theorem 5.1, we then omputethe He ke operator Tm. Finally, we ompute am(f) as

am(f) = ef (Tm).It is straightforward to he k that all these omputations an be done in time polynomial inthe length of the input.Remark 5.2. The proof shows that the Riemann hypothesis only needs to be assumedfor the ζ-fun tions of number elds that arise as elds of oe ients of primitive usp forms.6. Appli ations6.1. Counting points on modular urves. The ase k = 2 of Theorem 5.1 implies anew result on ounting points on modular urves over nite elds.Theorem 6.1. There exists a probabilisti algorithm that, given a squarefree positive in-teger n and a prime number p ∤ n, omputes the zeta fun tion of the modular urve X1(n)over Fp, and that runs in time polynomial in n and log p under the Riemann hypothesis forζ-fun tions of number elds.Proof. Let J1(n)Fp denote the Ja obian of X1(n)Fp . Let χ be the hara teristi polynomialof the Frobenius endomorphism of the l-adi Tate module TlJ1(n)Fp , where l is any primenumber dierent from p; then χ has integral oe ients and does not depend on the hoi eof l. Be ause of the well-known identity

ZX1(n)/Fp(t) =

χ∗(t)

(1 − t)(1 − pt),where χ∗(t) = tdeg χχ(1/t) is the re ipro al polynomial of χ, it su es to ompute χ.Let T1(n) denote the He ke algebra a ting on J1(n)Fp . Then Ql ⊗Zl

TlJ1(n)Fp is a freeQl⊗ZT1(n)-module of rank 2. By the Ei hlerShimura relation, the hara teristi polynomialof Frobp on it equals x2−Tpx+p〈p〉 ∈ T1(n)[x]. This implies that the hara teristi polynomialof Frobp viewed as a Ql-linear map equals

χ = NormT1(n)[x]/Z[x](x2 − Tpx+ p〈p〉) ∈ Z[x].Publi ations mathématiques de Besançon - 2011

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Peter Bruin 33To ompute the right-hand side, we use the fa t that the He ke algebras T(S2(Γ1(n))) andT1(n) are isomorphi . By Theorem 5.1, we an therefore ompute T1(n) and the matri esMTp and M〈p〉 of Tp and 〈p〉 with respe t to some Z-basis (b1, . . . , br) of T1(n). We interpret(b1, . . . , br) as a Z[x]-basis of T1(n)[x], and we ompute χ as the determinant of the matrixx2 · id − x ·MTp + p ·M〈p〉 with oe ients in Z[x].Corollary 6.2. There exists a probabilisti algorithm that, given a squarefree positive in-teger n and a prime power q oprime to n, omputes the number of rational points on X1(n)over the eld of q elements, and that runs in time polynomial in n and log q under the Riemannhypothesis for ζ-fun tions of number elds.6.2. Latti es. A parti ularly interesting family of modular forms onsists of θ-seriesasso iated to integral latti es. An integral latti e is a free Abelian group L of nite ranktogether with a symmetri , positive-denite, bilinear form

〈 , 〉 : L× L→ Z.We identify a latti e L with its image in the Eu lidean spa eLR = R ⊗Z L.The form 〈 , 〉 extends uniquely to an inner produ t 〈 , 〉R on LR. The dual latti e of L is

L∨ = v ∈ LR | 〈v, L〉 ⊆ Zequipped with the symmetri positive denite bilinear form 〈 , 〉∨ obtained by restri ting〈 , 〉R. The level of L is the exponent of the group L∨/L, i.e. the least positive integer csu h that cL∨ ⊆ L. It an be omputed as the least ommon denominator of the entries ofthe inverse of the matrix of 〈 , 〉 with respe t to some Z-basis of L.Let (L, 〈 , 〉) be an integral latti e of even rank k and level n. For every non-negativeinteger m we dene

rL(m) = #x ∈ L | 〈x, x〉 = m.The θ-series of L is the element of Z[[q]] dened byθL =

x∈L

q〈x,x〉

=

∞∑

m=0

rL(m)qm.This power series is the q-expansion of a modular form of weight k/2 for Γ1(4n). The latti eL is alled even if the integer 〈x, x〉 is even for all x ∈ L. If L is even, then the level 4n anbe repla ed by 2n; if both L and L∨ are even, then it an be repla ed by Γ1(n). For proofsof these results, we refer to Miyake [12, § 4.9.Couveignes, Edixhoven et al. [2, § 15.3 treat the following appli ation of their result on omputing oe ients of modular forms for SL2(Z). They take L equal to the Lee h latti e,whi h is the unique self-dual even latti e of rank 24. Its θ-series is a linear ombinationPubli ations mathématiques de Besançon - 2011

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34 Computing oe ients of modular formsof the Eisenstein series E12 and the dis riminant modular form ∆. The latter is the uniqueelement of S12(SL2(Z)) with a1(∆) = 1. Its q-expansion oe ients are given by Ramanujan'sτ -fun tion:

∆ = q

∞∏

m=1

(1 − qm)24

=

∞∑

m=1

τ(m)qm.As mentioned before, the oe ients of Eisenstein series an be omputed from the formulaein § 2.3. It is proved in [2 that given a positive integer m in fa tored form, the integerτ(m), and hen e the representation number rL(m), an be omputed deterministi ally intime polynomial in logm.The orresponding generalisation that is made possible by Theorem 1.1 is the following result.Theorem 6.3. Let n0 be a positive integer. There exists a probabilisti algorithm that,given an even positive integer k, a squarefree positive integer n1 oprime to n0, the representation numbers rL(0), . . . , rL(k/2 · d(Γ1(4n)) for a latti e L of even rank kand level n, where 4n = n0n1, and a positive integer m in fa tored form, omputes rL(m), and that runs in time polynomial in k, n1 and logm under the Riemannhypothesis for ζ-fun tions of number elds.Remark 6.4. Unfortunately, in general it is not lear how one an e iently ompute θLto su ient order, given only the matrix of 〈 , 〉 with respe t to some Z-basis of L.6.3. Sums of squares. Now onsider the latti e Zk, equipped with the standard bilinearform, so that the standard basis is orthonormal. Its θ-series is(13) θZk = θk,where θ is Ja obi's θ-series:

θ =∑

m∈Z

qm2= 1 + 2

∞∑

m=1

qm2.We let rk(m) denote the m-th oe ient of θZk , so that

rk(m) = #(x1, . . . , xk) ∈ Zk | x21 + · · · + x2

k = m.The problem of nding rk(m) is the lassi al problem of determining the number of waysin whi h m an be written as a sum of k squares. This question has a long and interestinghistory, whi h involves (among many others) Fermat, Legendre, Gauÿ, Ja obi, Eisenstein andLiouville. There is a large volume of literature devoted to this problem; we refer only toDi kson [7, Grosswald [8 and Milne [11.Publi ations mathématiques de Besançon - 2011

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Peter Bruin 35From now on we restri t to even values of k. This restri tion is imposed on us by the fa tthat θ is a modular form of weight 1/2, and our results on omputing oe ients of modularforms only hold for forms of integral weight.For k = 2, 4, 6, 8, 10, there exist formulae for rk(m). One set of su h formulae is the following:r2(m) = 4

d|m

ǫ(d),

r4(m) = 8∑

d|m

d− 32∑

d|(m/4)

d,

r6(m) = 16∑

d|m

ǫ(m/d)d2 − 4∑

d|m

ǫ(d)d2,

r8(m) = 16∑

d|m

d3 − 32∑

d|(m/2)

d3 + 256∑

d|(m/4)

d3,

r10(m) =4

5

d|m

ǫ(d)d4 +64

5

d|m

ǫ(m/d)d4 +8

5

z∈Z[√−1]

|z|2=m

z4.Here d runs over the positive divisors of m, m/2 or m/4; if m/2 or m/4 is not an integer,the orresponding sum is omitted. Furthermore, ǫ denotes the unique non-trivial Diri hlet hara ter modulo 4:ǫ(d) =

(−1

d

)

=

1 if d ≡ 1 mod 4,

−1 if d ≡ 3 mod 4,

0 if d ≡ 0 mod 2.One way to interpret the existen e of the above formulae is as follows. For k = 2, 4, 6, 8,the spa e Sk/2(Γ1(4)) is trivial; in other words, θZk ∈ Mk/2(Γ1(4)) is a linear ombination ofEisenstein series. Although S5(Γ1(4)) is non-trivial, it is spanned by a usp form with omplexmultipli ation, explaining the last term in the formula for r10(m).For k ≥ 12, it is true that various formulae have been proposed for rk(m), but it seems thatnone of these makes it possible to ompute rk(m) time polynomial in k and logm. This maybe understood, from our perspe tive, in light of the fa t that for every even k ≥ 12, thede omposition of θk as as a linear ombination of eigenforms ontains usp forms without omplex multipli ation. The latter fa t was proved re ently by I. Varma [16. No methodwas previously known for omputing the oe ients of su h usp forms in polynomial time.Using (13), we an qui kly ompute θZk to su ient order to determine it uniquely as anelement of Mk/2(Γ1(4)). The following result is therefore a spe ial ase (n0 = 4, n1 = 1) ofTheorem 6.3.Theorem 6.5. There exists a probabilisti algorithm that, given an even positive integer kand a positive integer m in fa tored form, omputes the number of representations of m asa sum of k squares of integers, and that runs in time polynomial in k and logm under theRiemann hypothesis for ζ-fun tions of number elds. Publi ations mathématiques de Besançon - 2011

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36 Computing oe ients of modular formsAs in Remark 1.3, without assuming the generalised Riemann hypothesis we an still provethat for xed k, the expe ted running time is polynomial in logm.Referen es[1 P. J. Bruin, Modular urves, Arakelov theory, algorithmi appli ations. Proefs hrift (Ph.D. the-sis), Universiteit Leiden, 2010. Available online: http://hdl.handle.net/1887/15915 .[2 J.-M. Couveignes and S. J. Edixhoven (with J. G. Bosman, R. S. de Jong and F. Merkl),Computational aspe ts of modular forms and Galois representations. Prin eton University Press, toappear. Preprint available on arXiv: math/0605244 .[3 P. Deligne, Formes modulaires et représentations l-adiques. Séminaire Bourbaki, 21e année(1968/1969), exposé 355. Le ture Notes in Mathemati s 179, 139172. Springer-Verlag, Berlin/Heidelberg/New York, 1971.[4 P. Deligne, La onje ture de Weil. I. Publi ations mathématiques de l'I.H.É.S. 43 (1973), 273307.[5 F.Diamond and J. Im, Modular forms and modular urves. In: V.Kumar Murty (editor), Semi-nar on Fermat's Last Theorem (Fields Institute for Resear h in Mathemati al S ien es, Toronto, ON,19931994), 39133. CMS Conferen e Pro eedings 17. Ameri an Mathemati al So iety, Providen e,RI, 1995.[6 F. Diamond and J. Shurman, A First Course in Modular Forms. Springer-Verlag,Berlin/Heidelberg/New York, 2005.[7 L. E. Di kson, History of the theory of numbers. Volume II: Diophantine analysis. CarnegieInstitute, Washington, D.C., 1920. Reprinted by Chelsea Publishing Co., New York, 1966.[8 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag,Berlin/Heidelberg/New York, 1985.[9 H. Iwanie and E. Kowalski, Analyti Number Theory . AMS Colloquium Publi ations 53.Ameri an Mathemati al So iety, Providen e, RI, 2004.[10 H. W. Lenstra, Jr. and C. Pomeran e, A rigorous time bound for fa toring integers. Journalof the Ameri an Mathemati al So iety 5 (1992), no. 3, 483516.[11 S. C. Milne, Innite families of exa t sums of squares formulas, Ja obi ellipti fun tions, on-tinued fra tions, and S hur fun tions. Ramanujan Journal 6 (2002), no. 1, 7149.[12 T. Miyake, Modular Forms. Springer-Verlag, Berlin/Heidelberg, 1989.[13 W. A. Stein,Modular Forms, a Computational Approa h. With an appendix by P. E.Gunnells.Ameri an Mathemati al So iety, Providen e, RI, 2007.[14 J. Sturm, On the ongruen e of modular forms. In: D. V. Chudnovsky, G. V. Chudnovsky,H. Cohn and M. B. Nathanson (editors), Number Theory (New York, 19841985). Le ture Notesin Mathemati s 1240, 275280. Springer-Verlag, Berlin/Heidelberg, 1987.[15 E. Ullmo, Hauteur de Faltings de quotients de J0(N), dis riminants d'algèbres de He ke et ongruen es entre formes modulaires. Ameri an Journal of Mathemati s 122 (2000), no. 1, 83115.[16 I. Varma, Sums of Squares, Modular Forms, and He ke Chara ters. Master's thesis, UniversiteitLeiden, 2010.1er o tobre 2010Peter Bruin, Département de Mathématiques d'Orsay, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay edex, Fran e • E-mail : Peter.Bruinmath.u-psud.frPubli ations mathématiques de Besançon - 2011

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APPLICATIONS OF THE ASYMPTOTIC LARGE SIEVEbyBrian ConreyAbstra t. In this note we des ribe some results of Conrey, Iwanie , and Soundararajan. Thedetails will appear elsewhere.Résumé. Dans ette note, nous présentons des résultats de Conrey, Iwanie et Soundara-rajan. Les détails seront publiés ultérieurement.

1. Introdu tionThe large sieve inequality, in its multipli ative form [H-B1, asserts that for any sequen e of omplex numbers an,S(~a,Q) :=

q≤Q

q

φ(q)

∑∗

χ mod q

N∑

n=1

anχ(n)∣

2 ≤ (Q2 + N)

N∑

n=1

|an|2.Many appli ations have followed from this basi tool, espe ially, in this form, appli ationshaving to do with primes in arithmeti progressions, or to do with statisti al averages ofDiri hlet L-fun tions. Generalizations of this basi tool have been given where the hara tersχ are repla ed by the oe ients of L-fun tions from a family. Thus, there are versions whi hinvolve only the real hara ters (see [H-B4, as well as versions with Fourier oe ients ofmodular forms (see [DI, [Iwa1, and [Iwa2) and many others. The large sieves reveal analmost orthogonality among the sets of oe ients.For ertain appli ations, it is desirable to have an asymptoti formula in pla e of the inequalityabove. We have developed su h an asymptoti formula, for ertain sequen es, and even haveformulas for

S(~a,Q) :=∑

q≤Q

q

φ(q)

∑∗

χ mod q

|L(1/2, χ)|2∣

N∑

n=1

anχ(n)∣

22000 Mathemati s Subje t Classi ation. 11M06, 11M26.Key words and phrases. Large sieve, Diri hlet L-fun tions, Generalized Riemann Hypothesis, momentsof L-fun tions.

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38 Appli ations of the asymptoti large sievefor L-fun tions of degrees 1, 2, and 3 and forS(~a,Q) :=

q≤Q

q

φ(q)

∑∗

χ mod q

|L(1/2, χ)|2k∣

N∑

n=1

anχ(n)∣

2provided that k times the degree of L is not too large.As a sample appli ation, we an show that, assuming the generalized Riemann Hypothesis,there are L-fun tions whi h have zeros spa ed loser than 0.366 times the average spa ing.Another appli ation is to riti al zeros of a family of twists of a xed L-fun tion. Let L(s) bean L-fun tion. We onsider the olle tionL(s, χ)where χ ranges over all primitive hara ters χ modulo q with q ≤ Q. We onsider all of thezeros of all of these L-fun tions up to a height log Q. If L is a degree one L-fun tion, then atleast 55% of all of these zeros are on the 1/2-line. If L is degree 2, then at least 35% are onthe riti al line, and if L is degree 3 then at least one-half of one per ent of the zeros are onthe riti al line.A third appli ation is to moments of Diri hlet L-fun tions. Here the best result is for thesixth moment of Diri hlet L-fun tions. We prove a formula, whi h ontains the full ninthdegree polynomial polynomial, and whi h agrees perfe tly with the onje tures motivated byrandom matrix theory. 2. The basi ideaThe essen e of S(~a,Q) is ∆ whi h is dened through

S(~a,Q) =∑

m,n≤X

ambn∆(m,n);thus∆(m,n) :=

q

W (q/Q)

φ(q)

∑∗

χ mod q

χ(m)χ(n).Here are some basi lemmas to get us started analyzing this ∆symbol.Lemma 2.1. If (mn, q) = 1, then∑∗

χ mod q

χ(m)χ(n) =∑

d|qd|(m−n)

φ(d)µ(q/d).Applying Lemma 2.1 we nd that∆(m,n) =

(cd,mn)=1d|m−n

W (cd/Q)µ(c)φ(d)

φ(cd).

Publi ations mathématiques de Besançon - 2011

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Brian Conrey 39Lemma 2.2. We haveφ(d)

φ(cd)=

1

φ(c)

a|ca|d

µ(a)

a.Thus,

∆(m,n) =∑

(acd,mn)=1ad|(m−n)

W (a2cd/Q)µ(a)µ(ac)

aφ(ac).Now we separate the diagonal terms from the non-diagonal ones.Proposition 2.3. We have

∆(m,m) = W (1)Qφ(m)

m

p

(

1 − 1

p2− 1

p3

)

p|m

(

1 − 1

p2− 1

p3

)−1

+ Oǫ((Qm)ǫ)Proof. We have∆(m,m) =

(acd,m)=1

µ(a)µ(ac)

aφ(ac)W

(

a2cd

Q

)

=1

2πi

(2)QsW (s)ζ(s)

p|m

(1 − 1ps )

(ac,m)=1

µ(a)µ(ac)

a1+2scsφ(ac)ds.The sums over a and c are absolutely onvergent for σ > 0 and W (s) is of rapid de ay in theverti al dire tion. Let ǫ > 0. We shift the path of integration to the ǫ-line and pi k up theresidue from the pole of ζ(s) at s = 1. Thus

∆(m,m) = W (1)Qφ(m)

m

(ac,m)=1

µ(a)µ(ac)

a3cφ(ac)+ O

(

(Qm)ǫ)

.The sum over a and c in the main term is=

p∤m

(

1 +1

p3(p − 1)− 1

p(p − 1)

)

=∏

p∤m

(

1 − 1

p2− 1

p3

)

.Now we shall assume that m 6= n. We introdu e a parameter C and split the sum over c in∆ so that we have ∆(m,n) = L(m,n) + U(m,n) where

L(m,n) =∑

(acd,mn)=1ad|m−n,c≤C

W (a2cd/Q)µ(a)µ(ac)

aφ(ac)andU(m,n) =

(acd,mn)=1ad|m−n,c>C

W (a2cd/Q)µ(a)µ(ac)

aφ(ac).Publi ations mathématiques de Besançon - 2011

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40 Appli ations of the asymptoti large sieveThe term U(m,n) is relatively easy to analyze. Basi ally we dete t the ongruen e m ≡n mod ad by going to hara ters modulo ad. Sin e c is large and acd ≈ Q it must be the asethat ad is small and so it is not too expensive to sum over all of the hara ters modulo ad.For the term L(m,n) with smaller c and larger ad we write |m − n| = ade where e is alledthe omplementary modulus and we then, after elimination of d from the rest of the sum,we have m ≡ n mod ae and dete t this ongruen e by hara ters modulo ae. The di ultyin doing this is largely te hni al; to eliminate the variable d everywhere by repla ing it by|m − n|/ae is ompli ated! In arrying out this pro edure for spe i sequen es an one oftenen ounters main terms arising from the prin ipal hara ters modulo ad in U and from theprin ipal hara ters modulo ae in L. The two largest of these main terms an el; the real o-diagonal ontribution omes from a se ondary main term in the prin ipal hara ters moduloae of L. This term ombines with the diagonal to form the nal main term.In the next two se tions we give further examples of pre ise theorems we an prove.3. The analogue of the Balasubramanian, Conrey, Heath-Brown onje tureTheorem 3.1. Suppose that W is smooth and supported on [1, 2]. Then

q

W( q

Q

)

χ mod q

|L(1/2, χ)|2χ(h)χ(k) = Ress=0

W (s + 2)π−s

Γ(

s+1/22

)2

Γ(

14

)2 Q2+s ζ(1 + 2s)

s

×φ(hk)

hk

(h, k)1+2s

(hk)1/2+s

p∤hk

(

1 − 1

p2+2s− 2

p2+

2

p3+2s+

1

p4− 1

p5+2s

)

+ Eh,kwhere∑

h,k≤N

λhλkEh,k ≪ (Q2−ǫ + (QN)1+ǫ)∑

h≤N

|(λ ∗ λ)h|2.Here λ ∗ λ is the Diri hlet onvolution of the sequen e λ with itself. Note that for a sequen eλh ≪ hǫ−1/2 the main term above is likely to be ≫ Q2 so that we have an asymptoti formulaprovided that N ≪ Q1−ǫ.The above is the analogue of a onje ture of Balasubramanian, Conrey, and Heath-Brown[BCH-B who proved that∫ T

0|ζ(1/2 + it)|2

N∑

n=1

λnn−it

2

dt = T∑

h,k≤N

λhλk

(

(h, k)√hk

(

logT (h, k)2

2πhk+ 2γ − 1

)

+ E(h, k)

)where∑

h,k≤N

λhλkEh,k ≪ (T 1−ǫ + T 1/2N)∑

n≤N

|λn|2.This formula gives an asymptoti formula provided that N ≪ T 1/2−ǫ. The authors onje turethat the asymptoti formula a tually holds provided that N ≪ T 1−ǫ (a onje ture whi h isequivalent to the Lindelöf hypothesis).Publi ations mathématiques de Besançon - 2011

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Brian Conrey 414. The sixth momentUnderstanding moments of families of L-fun tions has long been an important subje t withmany number theoreti appli ations. It is only in the last ten years that an understanding ofthe ner stru ture of moments has begun to emerge. The new vision began with the work ofKeating and Snaith, who realized that the distribution of the values of an L-fun tion, or familyof L-fun tions, an be modeled by hara teristi polynomials from lassi al ompa t groups,and to Katz and Sarnak for their realization that families of L-fun tions have symmetry typesasso iated with them that reveal whi h of the lassi al groups to use to model the family. See[KaSa, [KS1, [KS2, and [CF.Prior to these works, Conrey and Ghosh predi ted, on number theoreti grounds, that∫ T

0|ζ(1/2 + it)|6 dt ∼ 42

p

(

1 − 1

p

)4 (

1 +4

p+

1

p2

)

Tlog9 T

9!.The onje ture of Keating and Snaith agrees with this. Numeri ally, however, this onje tureis untestable. For example,

∫ 2350000

0|ζ(1/2 + it)|6 dt = 3317496016044.9 = 3.3 × 1012whereas

42∏

p

(

1 − 1

p

)4 (

1 +4

p+

1

p2

)

× 2350000 × (log 2350000)9

9!= 4.22 × 1011is nowhere near the predi tion. This situation has been re tied by the onje tures of[CFKRS whi h assert, for example, that for any ǫ > 0,

∫ T

0|ζ(1/2 + it)|6 dt =

∫ T

0P3

(

logt

)

dt + O(T 1/2+ǫ)where P3 is a polynomial of degree 9 whose exa t oe ients are spe ied as ompli atedinnite produ ts and series over primes, but whose approximate oe ients areP3(x) = 0.000005708x9 + 0.0004050x8 + 0.01107x7 + 0.1484x6

+1.0459x5 + 3.9843x4 + 8.6073x3 + 10.2743x2 + 6.5939x + 0.9165.For this polynomial, we have∫ 2350000

0P3

(

logt

)

dt = 3317437762612.4whi h agrees well with the numeri s.A proof of an asymptoti formula for the sixth moment of ζ(s) on the iti al line is ompletelyout of rea h of today's te hnology. However, we have proved an analogous formula for Diri hletL-fun tions suitably averaged. Our formula agrees exa tly with the onje ture of [CFKRSand so provides, we hope, a new glimpse into the me hani s of moments.Publi ations mathématiques de Besançon - 2011

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42 Appli ations of the asymptoti large sieveThe pre ise statement of our theorem on the sixth moment of Diri hlet L-fun tions is a little ompli ated to state and requires some set up. We invite the reader to look at the end of thepaper for a version of the theorem whi h shows the rst main term in the asymptoti formula.Let χ mod q be an even, primitive Diri hlet hara ter and letL(s, χ) =

∞∑

n=1

χ(n)

nsbe its asso iated L-fun tion. Su h an L-fun tion has an Euler produ t: L(s, χ) =∏

p(1 −χ(p)/ps)−1 and a fun tional equation

Λ(s, χ) := (q/π)(s−1/2)/2Γ(s/2)L(s, χ) = ǫχΛ(1 − s, χ)where ǫχ is a omplex number of absolute value 1. We prove an asymptoti formula, with apower savings, for a suitable average of the sixth power of the absolute value of these primitiveDiri hlet L-fun tions near the riti al point 1/2.An upper bound for su h an average follows from the large sieve inequality. Huxley used thisto prove that∑

q≤Q

∑∗

χ mod q

|L(1/2, χ)|6 ≪ Q2 log9 Qand∑

q≤Q

∑∗

χ mod q

|L(1/2, χ)|8 ≪ Q2 log16 Q.Here is a statement of those onje tures and the theorem of this paper. Let A and B be setsof omplex numbers with equal ardinality |A| = |B| = K. Suppose that |ℜα|, |ℜβ| ≤ 1/4 forα ∈ A, β ∈ B. These are the shifts. Let

ΛA,B(χ) :=∏

α∈A

Λ(1/2 + α,χ)∏

β∈B

Λ(1/2 + β, χ)

=( q

π

)δA,BGA,BLA,B(χ)whereLA,B(χ) :=

α∈A

L(1/2 + α,χ)∏

β∈B

L(1/2 + β, χ),

GA,B :=∏

α∈A

Γ

(

1/2 + α

2

)

β∈B

Γ

(

1/2 + β

2

)andδA,B =

1

2

α∈A

α +∑

β∈B

β

.Publi ations mathématiques de Besançon - 2011

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Brian Conrey 43Further, letZ(A,B) :=

α∈A

β∈B

ζ(1 + α + β)andA(A,B) :=

p

Bp(A,B)Zp(A,B)−1withZp(A,B) :=

α∈A

β∈B

ζp(1 + α + β)andζp(x) =

(

1 − 1

px

)−1

;alsoBp(A,B) :=

∫ 1

0

α∈A

zp,θ(1/2 + α)∏

β∈B

zp,−θ(1/2 + β) dθwith zp,θ(x) = 1/(1−e(θ)/px). The onditions on the real parts of elements of A and B ensurethat the Euler produ t for A onverges absolutely. Let Bq =∏

p|q Bp. LetQA,B(q) :=

S⊂AT⊂B

|S|=|T |

Q(S ∪ (−T ), T ∪ (−S); q)where S denotes the omplement of S in A, and by the set −S we mean −s : s ∈ S and,for any sets X and Y ,Q(X,Y ; q) =

( q

π

)δX,Y GX,YAZBq

(X,Y ).For example, if A = α1, α2, α3 and B = β1, β2, β3, thenQA,B(q) = Q(α1, α2, α3, β1, β2, β3, q) + Q(−β1, α2, α3, −α1, β2, β3, q)

+ · · · + Q(−β1,−β2,−β3, −α1,−α2,−α3, q),is a sum of 8 terms; the rst summand orresponds to S = T = φ so that X = A and Y = B;the se ond summand orresponds to S = α1 and T = β1, and so on. In general, QA,B(q)will have (

2KK

) summands.Now we state the onje ture.Conje ture 4.1. [CFKRS Assuming that the shifts α ∈ A, β ∈ B satisfy |ℜα|, |ℜβ| ≤1/4, and ℑα,ℑβ ≪ q1−ǫ, we onje ture that

χ mod q

ΛA,B(χ) =∑

χ mod q

QA,B(q)(1 + O(q−1/2+ǫ))Publi ations mathématiques de Besançon - 2011

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44 Appli ations of the asymptoti large sievewhere ∑ denotes a sum over even primitive hara ters.When |A| = |B| = 3 and all of the shifts are 0, the onje ture implies that, as q → ∞ with qnot ongruent to 2 modulo 4, we haveConje ture 4.2. [CFKRS1

φ(q)

χ mod q

|L(1

2, χ)|6 ∼ 42a3

p|q

(

1 − 1p

)5

(

1 + 4p + 1

p2

)

log9 q

9!.where φ(q) is the number of even primitive hara ters modulo q and

a3 =∏

p

(

1 − 1

p

)4 (

1 +4

p+

1

p2

)

.Note that a3 is the onstant that appears in the onje ture for the sixth moment of ζ. The`42' here played an important role in the dis overy by Keating and Snaith that momentsof L-fun tions ould be modeled by moments of hara teristi polynomials, see Beineke andHughes [BeHu for an a ount of this story.For a set A it is onvenient to let At be the set of translatesAt = α + t : α ∈ A.Note that (As1)s2 = As1+s2 .Theorem 4.3. Suppose that |A| = |B| = 3 and that α, β ≪ 1/ log Q, for α ∈ A, β ∈ B.Suppose that Ψ is smooth on [1, 2] and Φ(t) is an entire fun tion of t whi h de ays rapidly as

|t| → ∞ in any xed horizontal strip. Then∑

q

Ψ

(

q

Q

)∫ ∞

−∞Φ(t)

χ

ΛAit,B−it(χ) dt

=∑

q

Ψ

(

q

Q

)∫ ∞

−∞Φ(t)φ(q)QAit,B−it

(q) dt + O(Q7/4+ǫ).We ould equally well prove a theorem for odd primitive hara ters. The answer would besimilar with just the Gamma-fa tors hanged slightly to ree t the dieren e in the fun tionalequation for odd primitive Diri hlet L-fun tions. When the shifts are all 0, this dieren edisappears, in the leading order main term.Publi ations mathématiques de Besançon - 2011

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Brian Conrey 45Corollary 4.4. We have∑

q

∑∗

χ mod q

Ψ(q/Q)

∫ ∞

−∞Φ(t)|L(1/2 + it, χ)|6 dt

∼ 42a3

q

Ψ

(

q

Q

)

p|q

(

1 − 1p

)5

(

1 + 4p + 1

p2

)φ(q)log9 q

9!

∫ ∞

−∞Φ(t)|Γ((1/2 + it)/2)|6 dt

∼ 42a3(L)Q2 log9 Q

9!

∫ ∞

0Ψ(x)x dx

∫ ∞

−∞Φ(t)|Γ((1/2 + iy)/2)|6 dtwhere

a3(L) =∏

p

(

1 − 1

p

)5(

1 +5

p− 5

p2+

14

p3− 15

p4+

5

p5+

4

p6− 4

p7+

1

p8

)

.Thus, our theorem asserts that Conje tures 4.1 and 4.2 are true (but with a weaker errorterm) on average over q and a mild average over y.5. Con lusionThe asymptoti large sieve is potentially a very useful tool with many appli ations. In itsinitial in arnations, the proofs are fairly di ult, espe ially when omplex main terms haveto be unearthed. The authors are seeking good ways to present these new ideas, ways thatwill simplify some of these ompli ations.Referen es[BCH-B Balasubramanian, R.; Conrey, J. B.; Heath-Brown, D. R. Asymptoti mean square of theprodu t of the Riemann zeta-fun tion and a Diri hlet polynomial. J. Reine Angew. Math. 357 (1985),161181.[BeHu Beineke, J. and Hughes, C. Great moments of the Riemann zeta fun tion, in Bis uits of numbertheory. Edited by Arthur T. Benjamin and Ezra Brown. The Dol iani Mathemati al Expositions, 34.Mathemati al Asso iation of Ameri a, Washington, DC, 2009. xiv+311 pp.[BD Bombieri, E.; Davenport, H. Some inequalities involving trigonometri al polynomials. Ann.S uola Norm. Sup. Pisa (3) 23 (1969), 223241.[Con Conrey, J. B. The mean-square of Diri hlet L-fun tions, arXiv:0708.2699.[CF Conrey, J. B.; Farmer, D. W. Mean values of L-fun tions and symmetry. Internat. Math. Res.Noti es 2000, no. 17, 883908.[CFKRS Conrey, J. B.; Farmer, D. W.; Keating, J. P.; Rubinstein, M. O.; Snaith, N. C. Integralmoments of L-fun tions. Pro . London Math. So . (3) 91 (2005), no. 1, 33104.[CG Conrey, J. B.; Ghosh, A. A onje ture for the sixth power moment of the Riemann zeta-fun tion.Internat. Math. Res. Noti es 1998, no. 15, 775780.[CGo Conrey, J. B. ; Gonek, S. M. High moments of the Riemann zeta-fun tion. Duke Math. J. 107(2001), no. 3, 577604.[CIS Conrey, J. B., Iwanie , H., Soundararajan, K. The asymptoti large sieve. In preparation.Publi ations mathématiques de Besançon - 2011

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46 Appli ations of the asymptoti large sieve[DI Deshouillers, J.-M.; Iwanie , H. Kloosterman sums and Fourier oe ients of usp forms. Invent.Math. 70 (1982/83), no. 2, 219288.[H-B1 Heath-Brown, D. R. The fourth power moment of the Riemann zeta fun tion. Pro . LondonMath. So . (3) 38 (1979), no. 3, 385422.[H-B2 Heath-Brown, D. R. An asymptoti series for the mean value of Diri hlet L-fun tions. Com-ment. Math. Helv. 56 (1981), no. 1, 148161.[H-B3 Heath-Brown, D. R. The fourth power mean of Diri hlet's L-fun tions. Analysis 1 (1981), no.1, 2532.[H-B4 Heath-Brown, D. R. A mean value estimate for real hara ter sums. A ta Arith. 72 (1995),no. 3, 235275.[Hux Huxley, M. N. The large sieve inequality for algebrai number elds. II. Means of moments ofHe ke zeta-fun tions. Pro . London Math. So . (3) 21 (1970), 108-128.[Iwa1 Iwanie , Henryk Introdu tion to the spe tral theory of automorphi forms. Bibliote a de laRevista Matemati a Iberoameri ana. Revista Matemati a Iberoameri ana, Madrid, 1995.[Iwa2 Iwanie , Henryk Topi s in lassi al automorphi forms. Graduate Studies in Mathemati s, 17.Ameri an Mathemati al So iety, Providen e, RI, 1997.[KaSa Katz, Ni holas M.; Sarnak, Peter Random matri es, Frobenius eigenvalues, and monodromy.Ameri an Mathemati al So iety Colloquium Publi ations, 45. Ameri an Mathemati al So iety, Prov-iden e, RI, 1999.[KS1 Keating, J. P.; Snaith, N. C. Random matrix theory and ζ(1/2 + it). Comm. Math. Phys. 214(2000), no. 1, 5789.[KS2 Keating, J. P.; Snaith, N. C. Random matrix theory and L-fun tions at s = 1/2. Comm. Math.Phys. 214 (2000), no. 1, 91110.[Sou Soundararajan, K. The fourth moment of Diri hlet L-fun tions, Analyti number theory, 239-246, Clay Math. Pro ., 7, Amer. Math. So ., Providen e, RI, 2007; arxiv math.NT/0507150.[Sou1 Soundararajan, K. Moments of the Riemann zeta-fun tion, Ann. of Math. (2) 170 (2009), no.2, 981-993.; arxiv math.NT/0612106.[You Young, Matthew. The fourth moment of Diri hlet L-fun tions, to appear, Ann. of Math.;arxivmath.NT/0610335.Lundi 31 janvier 2011Brian Conrey, Ameri an Institute of Mathemati s, 360 Portage Ave Palo Alto, CA 94306-2244Department of Mathemati s University Walk, Clifton, Bristol BS8 1TW, U.K.E-mail : onreyaimath.org

Publi ations mathématiques de Besançon - 2011

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ASYMPTOTIC METHODS IN NUMBER THEORY ANDALGEBRAIC GEOMETRYbyPhilippe Leba que & Alexey ZykinAbstra t. The paper is a survey of re ent developments in the asymptoti theory of globalelds and varieties over them. First, we give a detailed motivated introdu tion to the asymptoti theory of global elds whi h is already well shaped as a subje t. Se ond, we treat in a moresket hy way the higher dimensional theory where mu h less is known and many new resear hdire tions are available.Résumé. Cet arti le est un survol des développements ré ents dans la théorie asymptotiquedes orps globaux et des variétés algébriques dénies sur les orps globaux. Dans un premiertemps, nous donnons une introdu tion détaillée et motivée à la théorie asymptotique des orpsglobaux, théorie déjà bien établie. Puis nous aborderons plus rapidement la théorie asymptotiqueen dimension supérieure où peu de hoses sont onnues et où bien des dire tions de re her hesont ouvertes.1. Introdu tion: the origin of the asymptoti theory of global eldsThe goal of this arti le is to give a survey of asymptoti methods in number theory andalgebrai geometry developed in the last de ades. The problems that are treated by theasymptoti theory of global elds (that is number elds or fun tion elds) and varieties overthem are quite diverse in nature. However, they are onne ted by the use of zeta fun tions,whi h play the key role in the asymptoti theory.We begin by a very well known problem whi h lies at the origin of the asymptoti theoryof global elds. Let Fr be the nite eld with r elements. For a smooth proje tive urve Cover Fr we let Nr(C) be the number of Fr-point on C. We denote by g(C) be the genus of C.2000 Mathemati s Subje t Classi ation. 11R42,11R29,11G40.Key words and phrases. Towers of global elds, L-fun tions in family, BrauerSiegel theorem.The rst author was partially supported by EPSRC grant EP/E049109 Two dimensional adeli analysis". These ond author was partially supported by Dmitry Zimin's Dynasty" foundation, by AG Laboratory SU-HSE,RF government grant, ag. 11.G34.31.0023, by the grants RFBR 10-01-93110-CNRSa, RFBR 11-01-00393-a,by the grant of the Ministry of Edu ation and S ien e of Russia No. 2010-1.3.1-111-017-029. The individualresear h proje t N 10-01-0054 Asymptoti properties of zeta fun tions" was arried out with the support ofthe program S ienti fund of HSE".

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48 Asymptoti methods in number theory and algebrai geometryThe problem onsists of nding the maximum Nr(g) of the numbers Nr(C) over all smoothproje tive urves of genus g over Fr :

Nr(g) = maxg(C)=g

Nr(C).The rst upper bound was dis overed by André Weil in 1940s as a dire t onsequen e of hisproof of the Riemann hypothesis for urves over nite elds. He showed that Nr(C) satisesthe inequalityNr(C) ≤ r + 1 + 2g

√r.Weil bound though extremely useful in many appli ations is far from being optimal. Adramati sear h for the improvements of this bound and for the examples giving lower boundson Nr(g) has begun in 1980s with the dis overy of Goppa that urves over nite elds withmany points an be used to onstru t good error- orre ting odes. To show how importantthe developments in this area were it su es to mention the names of some mathemati ianswho turned their attention to these questions: J.-P. Serre, V. Drinfeld, Y. Ihara, H. Stark, R.S hoof, M. Tsfasman, S. Vl duµ, G. van der Geer, K. Lauter, H. Sti htenoth, A. Gar ia, et .As suggested in [Ser85 by J.-P. Serre the ases when g is small and that when g is largerequire ompletely dierent treatment. That is the latter ase whi h interests us in this arti le.The rst major result in this dire tion was the following theorem of V. Drinfeld and S. Vl duµ[DV:Theorem 1.1 (DrinfeldVl duµ). For any family of smooth proje tive urves Ci over

Fr of growing genus we havelim sup

i→∞

Nr(Ci)

g(Ci)≤

√r − 1.Moreover, in the ase, when r is a square this bound turns out to be optimal. The familiesof urves, attaining this bound are onstru ted in many dierent ways: modular urves,Drinfeld modular urves, expli it iterated onstru tions, et . We refer the reader to se tion 4for more details. This result, signi antly improved and then reinterpreted in terms of limitzeta fun tions by M. Tsfasman and S. Vl duµ, lies at the very base of the asymptoti theoryof global elds. We will dis uss all this in detail in se tion 2. It is also possible to extendthe DrinfeldVl duµ inequalities to the ase of higher dimensional varieties. This serves as akeystone in the onstru tion of the higher dimensional asymptoti theory (see se tion 5).We will now turn our attention to yet another sour e of development of the asymptoti theory,this time in the ase of number elds. Let K be an algebrai number eld, that is a niteextension of Q. We denote by nK = [K : Q] its degree, and by DK its dis riminant. Animportant question (both on its own a ount and due to its appli ations in various domainsof number theory, arithmeti geometry and theory of sphere pa kings) is to know the rate ofgrows of dis riminants of number elds. The rst bound onDK was obtained by H. Minkowskyusing the geometry of numbers. This bound was improved more than half a entury later byH. Stark, J.-P. Serre and A. Odlyzko ([Sta74, [Ser75, [Odl76, [Odl90) who used analyti methods involving zeta fun tions. The bounds they prove are as follows:Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 49Theorem 1.2 (Odlyzko). For a family of number elds Ki we havelog |DKi

| ≥ A · r1(Ki) + 2B · r2(Ki) + o(nKi),where r1(Ki) and r2(Ki) are respe tively the number of real and omplex pla es of Ki. Un on-ditionally, we an take A = log(4π) + γ + 1 ≈ 60.8, B = log(4π) + γ ≈ 22.3, and, assumingthe generalized Riemann Hypothesis (GRH), one an take, A = log(8π)+ γ+ π

2 ≈ 215.3, B =

log(8π) + γ ≈ 44.7, where γ = 0.577 is Euler's gamma onstant.The fa t that GRH drasti ally improves the results is omnipresent in the asymptoti theoryof global elds. Fortunately, GRH is known for zeta fun tions of urves over nite elds (Weilbounds) and, more generally, of varieties over nite elds (Deligne's theorem), whi h allowsto have both stronger results and simpler proofs in the ase of positive hara teristi .M. Tsfasman and S. Vl duµ managed to generalize the above inequalities taking into a ountthe ontribution of nite pla es of the elds. In fa t, the restri tion of the so- alled basi inequality proven by M. Tsfasman and S. Vl duµ to innite primes gives us the inequalitiesof OdlyzkoSerre. If we restri t the basi inequality to nite pla es we obtain an analogue ofthe generalized DrinfeldVl duµ inequality in the ase of number elds. The reader will ndmore information on this in the next se tion of the paper.The last, but not least, problem that led to the development of the asymptoti theory of globalelds and varieties over them was the BrauerSiegel theorem. Let hK denote the lass numberof a number eld K and let RK be its regulator. The lassi al BrauerSiegel theorem, provenby Siegel ([Sie) in the ase of quadrati elds and by Brauer ([Bra) in general des ribesthe behaviour of the produ t hKRK in families of number elds. The initial motivation for itwas a onje ture of Gauss on imaginary quadrati elds, however it has got many importantappli ations elsewhere. The theorem an be stated as follows:Theorem 1.3 (BrauerSiegel). For a family of number elds Ki we havelimi→∞

log(hKiRKi

)

log√

|DKi|

= 1provided the family satises two onditions:(i) limi→∞

nKi

gKi

= 0;(ii) either GRH holds, or all the elds Ki are normal over Q.It is possible to remove the rst and relax the se ond onditions of the theorem. The rst steptowards it was made by Y. Ihara in [Iha83 who onsidered families of unramied numberelds. A omplete answer (at least modulo GRH) was given by M. Tsfasman and S. Vl duµin [TV02 who showed how to treat this problem in the framework of the asymptoti theoryof number elds, in parti ular using the on ept of limit zeta fun tions. The orrespondingquestion for urves over nite elds is also of great interest sin e it des ribes the asymptoti behaviour of the number of rational points on Ja obians of urves over nite elds. All thiswill be dis ussed in detail in the se tion 3. Publi ations mathématiques de Besançon - 2011

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50 Asymptoti methods in number theory and algebrai geometryIn our introdu tion we mostly onsidered the one dimensional ase of number elds or fun tionelds. Here the theory is best developed. However, there is quite a number of results and onje tures for higher dimensional varieties with parti ularly ni e arithmeti al appli ations.Some of the results in this a tively developing area are dis ussed in se tion 5.Let us nally say that, despite of the fa t that the theory of error orre ting odes and thetheory of sphere pa kings are just briey mentioned in our introdu tion their role in the reation of the asymptoti theory of global elds is fundamental. Indeed many questionssome of whi h were mentioned here (maximal number of points on urves, growth of thedis riminants, et .) re eived parti ular attention due to their relation to error- orre ting odes or sphere pa kings.2. Basi on epts and results. TsfasmanVl duµ invariants of innite globaleldsMany authors onsidered the behaviour of arithmeti data (de omposition of primes, genus,root dis riminant, lass number, regulator et .) in families of global elds. Tsfasman andVl duµ laid the foundation for the asymptoti theory of global elds in order not to onsiderelds in a family, but the limit obje t (say, a limit zeta-fun tion) that would en ode theinformation on erning the asymptoti s of the initial arithmeti data.In this se tion we introdu e some denitions and give basi properties of families of globalelds.2.1. TsfasmanVl duµ invariants. Arguments and proofs for the results from thissubse tion an be found in [TV02. Let us rst dene the obje ts we are to work with. Letr be a power of a prime p, and let Fr denote the algebrai losure of Fr.Denition 2.1. A family of global elds is a sequen e K = Knn∈N su h that:1. Either all the Kn are nite extensions of Q or all the Kn are nite extensions of Fr(t)with Fr ∩Kn = Fr.2. if i 6= j, Ki is not isomorphi to Kj .A tower of global elds is a family satisfying in addition Kn ⊂ Kn+1 for every n ∈ N. Aninnite global (resp. number, resp. fun tion) eld is the limit of a tower of global (resp.number, resp. fun tion) elds, i.e. it is the union ∞

n=1Kn.Denition 2.2. The genus gK of a fun tion eld is the genus of the orresponding smoothproje tive urve. We dene the genus of a number eld K as gK = log

|DK |, where DK isthe dis riminant of K.As there are (up to an isomorphism) only nitely many global elds with genus smaller thana xed real number g, we have the following proposition.Proposition 2.3. For any family Ki of global elds the genus gKi→ +∞.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 51Thus, in the number elds ase, any innite algebrai extension of Q is an innite numbereld, whereas in the fun tion elds ase, we require the innite algebrai extension of Fr(t)to ontain a sequen e of fun tion elds with genus going to innity.Let us now dene the so- alled TsfasmanVl duµ invariants of a family of global elds.Throughout the paper, we use the a ronyms NF and FF for the number eld and the fun tioneld ases respe tively. As before, the GRH indi ation means that we assume the generalizedRiemann Hypothesis for Dedekind zeta-fun tions.First we introdu e some notation to be used throughout the paper:Q the eld Q (NF), Fr(t) (FF);nK [K : Q];

DK dis riminant of K (NF);gK the genus of K (FF ), the genus of K equal to log

|DK | (NF );Plf (K) the set of nite pla es of K;Np the norm of a pla e p ∈ Plf (K);deg p logr Np (FF );Φq(K) the number of pla es of K of norm q;ΦR(K) the number of real pla es of K (NF);ΦC(K) the number of omplex pla es of K (NF).We onsider the set of possible indi es for the Φq,

A =

R,C, pk | p prime, k ∈ Z>0

(NF )

rk | k ∈ Z>0

(FF ),and Af its subset of nite parameters

pk | p prime, k ∈ Z>0

.Denition 2.4. We say that a family K = Ki of global elds is asymptoti ally exa tif the following limit exists for any q ∈ A :

φq := limi→+∞

Φq(Ki)

gKi

.It is said to be asymptoti ally good if in addition one of the φq is nonzero, and asymptoti allybad otherwise. The numbers φq are alled the TsfasmanVl duµ invariants of the family K.This denition has two origins. The rst one is the information theory sin e the familiesgiving good algebrai geometri odes are those for whi h φr exists and is big. The se ondone is more te hni al and an be seen through Weil's expli it formulae. For onvenien e wealso put φ∞ = limnKi

gKi

= φR + 2φC.Being asymptoti ally exa t is not a restri tive ondition. To be pre ise:Proposition 2.5. 1. Any family of global elds ontains an asymptoti ally exa t sub-family.2. Any tower of global elds is asymptoti ally exa t and the φq's depend only on the limit.Publi ations mathématiques de Besançon - 2011

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52 Asymptoti methods in number theory and algebrai geometryWe an thus dene the TsfasmanVl duµ invariants of an innite global elds K as the invari-ants of any tower having limit K. From now on, we only onsider asymptoti ally exa t families,sin e they provide natural framework for asymptoti onsiderations. One of the problems ofthe asymptoti theory is to understand the set of possible φq. In the next propositions wedes ribe some the general properties of the φq. Let us start with the basi inequalities:Theorem 2.6 (TsfasmanVl duµ). For any asymptoti ally exa t family of global elds,the following inequalities hold:(NF −GRH)

q

φq log q√q − 1

+ (log√

8π +π

4+γ

2)φR + (log 8π + γ)φC ≤ 1,

(NF )∑

q

φq log q

q − 1+ (log 2

√π +

γ

2)φR + (log 2π + γ)φC ≤ 1,

(FF )

∞∑

m=1

mφrm

rm2 − 1

≤ 1,where γ is the Euler onstant.This result is entral in what follows. For instan e, it is used to show the onvergen e ofthe limit zeta-fun tion asso iated to the family. It is proven using the Weil expli it formulae,the ee tive Chebotarev density theorem for number elds and the Riemann hypothesis forfun tion elds.In the ase of towers of number elds (and of fun tion elds if we onsider suitable quantities),the degree of the extension gives an upper bound for the number of pla es above a primenumber p:Proposition 2.7. For an asymptoti ally exa t family of number elds and any prime num-ber p the following inequality holds:+∞∑

m=1

mφpm ≤ φR + 2φC.Let us nally dene the de ien y δK of an asymptoti ally exa t family K = Ki of globalelds as the dieren e between the two sides of the basi inequalities under GRH:(NF ) δK = 1 −

q

φq log q√q − 1

− (log√

8π +π

4+γ

2)φR − (log 8π + γ)φCand

(FF ) δK = 1 −∞

m=1

mφrm

rm2 − 1

.A remarkable fa t is that the de ien y of innite global elds is in reasing with respe t tothe in lusion (see [Leb10): K ⊂ L implies δK ≤ δL. One knows that elds of zero de ien yexist in the fun tion elds ase ( .f. se tion 4). Su h innite global elds are alled optimal,and they are of parti ular interest for the information theory.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 532.2. Rami ation, prime de omposition and invariants. The pre ise statementsand proofs of the results from this subse tion an be found in [GSR and [Leb10. TheTsfasmanVl duµ invariants of innite global elds ontain information on the rami ationand the de omposition of pla es in these elds. Indeed, one sees from Hurwitz genus formulathat any nitely ramied and tamely ramied tower of number elds is asymptoti ally good(be ause it has bounded root dis riminant). For fun tion elds, we have to ask in additionfor the existen e of a split pla e. It is not ex luded that there exists an asymptoti ally goodinnite global eld with innitely many ramied pla es and no split pla e, but no exampleshave been found so far. In the ase of fun tion elds, A. Gar ia and H. Sti htenoth provideda widely ramied optimal tower and an everywhere ramied tower of fun tion elds withbounded g/n is onstru ted in [DPZ. Unfortunately, we do not know anything similar fornumber elds.In general, we expe t asymptoti ally good towers to have very little rami ation and somesplit pla es. The next question, rst raised by Y. Ihara, is how many pla es split ompletelyin a tower K of global eld. It follows from the Chebotarev density theorem that the set of ompletely split pla es has in general a zero analyti density, that islim

s→1+

p∈D Np−s

p∈Plf (Q) Np−s= 0,where D is the set of pla es of Q that split ompletely in K/Q. In the ase of asymptoti allygood elds, ∑

p∈D

Np−1 is even bounded. However, in the ase of asymptoti ally bad elds, thenumerator an have an innite limit whereas the rami ation lo us is very small (but innite).We refer the reader to [Leb10 for a more detailed treatment of the above questions.3. Generalized BrauerSiegel theorem and limit zeta-fun tions3.1. Generalizations of the BrauerSiegel theorem. Now we turn our attention tothe BrauerSiegel theorem. The in-depth study of mathemati al tools involved in it leads toan important notion of limit zeta fun tions whi h plays a key role in the study of asymptoti problems.While looking at the statement of the BrauerSiegel theorem (theorem 1.3) one immediatelyasks a question whether the two onditions present in it are indeed ne essary. It is a rightguess that the se ond ondition involving normality is te hni al in its nature (though gettingrid of it would be a breakthrough in the analyti number theory sin e it is related to theso- alled Siegel zeroes of zeta-fun tions the real zeroes whi h lie abnormally lose to s = 1;of ourse, presumably they do not exist). The se ond ondition nK/ log√

|DK | → 0 looksmu h tri kier. Using the inequalities from proposition 2.7 it is immediate that this onditionis equivalent to the fa t that the family we onsider is asymptoti ally bad.A fundamental theorem of M. Tsfasman and S. Vl duµ from [TV02 allows both to treat theasymptoti ally good ase of the BrauerSiegel theorem and to relax the se ond ondition.Publi ations mathématiques de Besançon - 2011

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54 Asymptoti methods in number theory and algebrai geometryWe formulate it together with a omplementary result by A. Zykin [Zyk05 whi h relaxesthe se ond ondition in the asymptoti ally bad ase. Before stating the result we give thefollowing denition:Denition 3.1. We say that a number eld K is almost normal if there exists a towerK = Kn ⊃ · · · ⊃ K1 ⊃ K0 = Q, where ea h step Ki/Ki−1 is normal.Theorem 3.2 (TsfasmanVl duµZykin). Assume that for an asymptoti ally exa tfamily of number elds Ki either GRH holds or all the elds Ki are almost normal. Thenwe have:

limi→∞

log(hKiRKi

)

gKi

= 1 +∑

q

φq logq

q − 1− φR log 2 − φC log 2π,the sum being taken over all prime powers q.For an asymptoti ally bad family of number elds we have φR = 0 and φC = 0 as well as φq = 0for all prime powers q, so the on lusion of the theorem takes the form of that of the lassi alBrauerSiegel theorem. However, there are examples of families of number elds where theright hand side of the equality in the theorem is either stri tly less or stri tly greater thanone (see [TV02). Let us mention one parti ularly ni e orollary of the generalized BrauerSiegel theorem due to M. Tsfasman and S. Vl duµ: a bound on the regulators that improvesZimmert's bound (see [Zim, his bound an be written as lim inf

log RKi

gKi

≥ (log 2 + γ)φR +

2γφC).Theorem 3.3 (TsfasmanVl duµ). For a family of almost normal number elds Ki(or any number elds under the assumption of GRH) we havelim inf

logRKi

gKi

≥ (log√πe+ γ/2)φR + (log 2 + γ)φC.The proof of this bound is far from being trivial, it an be found in [TV02.The fun tion eld version of the BrauerSiegel theorem is both easier to prove and requiresno supplementary onditions (like normality or GRH). In fa t, it was obtained before the orresponding theorem for number elds and allowed to guess what the result for numberelds should be (for a proof see [Tsf92 or [TV97).Theorem 3.4 (TsfasmanVl duµ). For an asymptoti ally exa t family of smooth pro-je tive urves Xi over a nite eld Fr we have:

limi→∞

log hi

gi= log r +

∞∑

f=1

φrf logrf

rf − 1,where hi = h(Xi) = |(JacXi)(Fr)| is the ardinality of the Ja obian of Xi over Fr.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 55Let κK = Ress=1

ζK(s) be the residue of the Dedekind zeta fun tion ζK(s) =∏

q(1 − q−s)−Φq(K)of the eld K at s = 1. Using the residue formula (see [Lan94, Chapter VIII and [TVN,Chapter III)

κK =2ΦR(K)(2π)ΦC(K)hKRK

wK

|DK |(NF ase);

κK =hKr

g

(r − 1) log r(FF ase)(here wK is the number of roots of unity in K) one an see that the question about thebehaviour of the ratio from the BrauerSiegel theorem is redu ed to the orresponding questionfor κK . To put it into a more general framework, we rst seek an interpretation of thearithmeti quantities we would like to study in terms of spe ial values of ertain zeta fun tions,then we study the behaviour of these spe ial values in families using analyti methods. Wewill see in se tion 5 another appli ations of this prin iple. One also noti es that this redu tionstep explains the appearan e of the GRH in the statement of the BrauerSiegel theorem.Let us formulate yet another version of the generalized BrauerSiegel theorem proven byLeba que in [Leb07, Theorem 7. It has the advantage of being expli it with respe t to theerror terms, thus giving information about the BrauerSiegel ratio on the nite level.Theorem 3.5 (Leba que). Let K be a global eld. Then(i) in the fun tion eld ase

log(κK log r) =N

f=1

Φrf logrf

rf − 1− logN − γ +O

( gK

NrN/2

)

+O

(

1

N

)

;(ii) in the number eld ase assuming GRHlog κK =

q≤x

Φq logq

q − 1− log log x− γ +O

(

nK log x√x

)

+O

(

gK√x

)

,where γ = 0.577 . . . is the Euler onstant. The onstants in O are absolute and ee tively omputable (and, in fa t, not very big).This theorem an also be regarded as a generalization of the Mertens theorem (see [Leb07).A slight improvement of the error term (as before, assuming GRH) was obtained in [LZ. Anun onditional number eld version of this result is also available but is a little more di ultto state ([Leb07, Theorem 6). We should also note that Leba que's approa h leads to aunied proof of the asymptoti ally bad and asymptoti ally good ases of theorem 3.2 with orwithout the assumption of GRH.3.2. Limit zeta-fun tions. For the moment the asymptoti theory of global elds lookslike a olle tion of similar but not dire tly related results. The situation is laried immenselyby means of the introdu tion of limit zeta fun tions. Publi ations mathématiques de Besançon - 2011

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56 Asymptoti methods in number theory and algebrai geometryDenition 3.6. The limit zeta fun tion of an asymptoti ally exa t family of global eldsK = Ki is dened as

ζK(s) =∏

q

(1 − q−s)−φq(K),the produ t being taken over all prime powers in the number eld ase and over prime powersof the form q = rf in the ase of urves over Fr.The basi inequalities from theorem 2.6 give the onvergen e of the above innite produ tfor Re s ≥ 12 with the assumption of GRH and for Re s ≥ 1 without it (in parti ular, in thefun tion eld ase the innite produ t onverges for Re s ≥ 1

2). In fa t, the basi inequalitiesthemselves an be restated in terms of the values of limit zeta fun tions. To formulate themwe introdu e the ompleted limit zeta fun tion:ζK(s) = es2−φRπ−sφR/2(2π)−sφCΓ

(s

2

)φR

Γ(s)φCζK(s) (NF ase);ζK(s) = rsζK(s) (FF ase).Let ξK(s) = ζ ′K(s)/ζK(s) be the logarithmi derivative of the ompleted limit zeta fun tion.Then the basi inequalities from se tion 2 take the following form:Theorem 3.7 (Basi inequalities). For an asymptoti ally exa t family of global elds

K = Ki we have ξK(12) ≥ 0 in the fun tion eld ase and assuming GRH in the numbereld ase and ξK(1) ≥ 0 without the assumption of GRH.Let us give an interesting interpretation of the de ien y in terms of the distribution of zeroesof zeta fun tions on the riti al line. In fa t, the results we are going to state are interestingon their own. To a global eld K we asso iate the ounting measure ∆K = 1

gK

ρδt(ρ), where

t(ρ) = Im ρ in the number eld ase and t(ρ) = 1log r Im ρ in the fun tion ase; the sum istaken over all zeroes ρ of ζK(s) in the number eld ase and over all zeroes ρ of ζK(s) with

t(ρ) ∈ (−π, π] in the fun tion eld ase (in the ase of fun tion elds ζK(s) is periodi withthe period equal to 2π/ log r), δt is the Dira (atomi ) measure at t. Thus we get a measure onR in the number eld ase and on R/Z in the fun tion eld ase. The asymptoti behaviourof ∆K was rst onsidered by Lang [Lan71 in the asymptoti ally bad ase. The followingresult is proven in [TV02, Theorem 5.2 and [TV97, Theorem 2.1.Theorem 3.8 (TsfasmanVl duµ). For an asymptoti ally exa t family of global eldsK = Ki, assuming GRH, the limit lim

i→∞∆Ki

exists in an appropriate spa e of measures (tobe pre ise, in the spa e of measures of slow growth on R in the NF ase,and in the spa e ofmeasures on R/Z in the FF ase). Moreover, the limit is a measure with ontinuous densityMK(t) = Re ξK

(

12 + it

)

.Of ourse, the expression for MK(t) an be written expli itly using the invariants φq. Letus note two important orollaries of the theorem. First, we get an interpretation for thePubli ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 57de ien y δK = ξK(

12

)

= MK(0) as the asymptoti number of zeroes of ζKi(s) a umulatingat s = 1

2 . Se ond, the theorem shows that for any family of number elds zeroes of theirzeta-fun tions get arbitrarily lose to s = 12 (and, in a sense, we even know the rate at whi hzeroes of ζKi

(s) approa h to this point).3.3. Limit zeta-fun tions and BrauerSiegel type results. Let us turn our at-tention to the BrauerSiegel type results. The formulae from theorems 3.2 and 3.4 an berewritten as limi→∞

log κKi

gKi

= log ζK(1). Furthermore, using the absolute and uniform onver-gen e of innite produ ts for zeta fun tions for Re s > 1, Tsfasman and Vl duµ prove in[TV02, Proposition 4.2 that for Re s > 1 the equality limi→∞

log ζKi(s)

gKi

= log ζK(s) holds. Infa t, this equality remains valid for Re s < 1 (at least if we assume GRH in the number eld ase). The proof of the next theorem an be found in [Zy10 in the number eld ase and in[Zyk11 in the fun tion eld ase (where the same problem is treated in a broader ontext).Theorem 3.9 (Zykin). For an asymptoti ally exa t family of global elds K = Ki forRe s > 1

2 we havelimi→∞

log((s− 1)ζKi(s))

gKi

= log ζK(s) (NF ase assuming GRH);limi→∞

log((rs − 1)ζKi(s))

gKi

= log ζK(s) (FF ase).The onvergen e is uniform on ompa t subsets of the half-plane s | Re s > 12.The ase s = 1 of theorem 3.9 is equivalent to the BrauerSiegel theorem and urrent te h-niques does not allow to treat it in full generality without the assumption of GRH. Thusgetting un onditional results similar to theorem 3.9 looks ina essible at the moment. Theanalogue of the above result for s = 1

2 is onsiderably weaker and one has only an upperbound:Theorem 3.10 (Zykin). Let ρKibe the rst non-zero oe ient in the Taylor series ex-pansion of ζKi

(s) at s = 12 , i. e. ζKi

(s) = ρKi

(

s− 12

)rKi +o((

s− 12

)rKi)

. Then in the fun tioneld ase or in the number eld ase assuming that GRH is true, for any asymptoti ally exa tfamily of global elds K = Ki the following inequality holds:lim sup

i→∞

log |ρKi|

gKi

≤ log ζK

(

1

2

)

.The interest in the study of the asymptoti behaviour of zeta fun tions at s = 12 is partlymotivated by the orresponding problem for L-fun tions of ellipti urves over global elds,where this value is related to deep arithmeti invariants of the ellipti urves via the Bir hSwinnerton-Dyer onje ture. We refer the reader to se tion 5 for more details. The questionwhether the equality holds in theorem 3.10 is rather deli ate. It is related to the so alledlow-lying zeroes of zeta fun tions, that is the zeroes of ζK(s) having small imaginary partPubli ations mathématiques de Besançon - 2011

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58 Asymptoti methods in number theory and algebrai geometry ompared to gK . It might well happen that the equality limi→∞

log |ρKi|

gKi

= log ζK(12) does nothold for all asymptoti ally exa t families K = Ki sin e the behaviour of low-lying zeroesis known to be rather random. Nevertheless, it might hold for most families (whatever itmight mean).To illustrate how hard the problem may be, let us remark that Iwanie and Sarnak studieda similar question for the entral values of L-fun tions of Diri hlet hara ters [IS99 andmodular forms [IS00. They manage to prove that there exists a positive proportion ofDiri hlet hara ters (modular forms) for whi h the logarithm of the entral value of the orresponding L-fun tions divided by the logarithm of the analyti ondu tor tends to zero.The te hniques of the evaluation of mollied moments used in these papers are rather involved.We also note that, to our knowledge, there has been no investigation of low-lying zeroes of

L-fun tions of growing degree. It seems that the analogous problem in the fun tion eld asehas neither been very well studied.Let us indi ate that the orresponding question for the logarithmi derivatives of zeta fun tionshas a negative answer. Indeed, the fun tional equation implies that limi→∞

ζ′Ki

(1/2)

ζKi(1/2) = 1 for anyfamily of fun tion elds Ki. However, the logarithmi derivative of the limit zeta fun tion

ζK(s) at s = 12 equals one only for asymptoti ally optimal families ( .f. theorem 3.7).As a orollary of theorem 3.9 one an obtain a result on the asymptoti behaviour of EulerKrone ker onstants.Denition 3.11. The EulerKrone ker onstant of a global eld K is dened as γK =

c0(K)c−1(K) , where ζK(s) = c−1(K)(s− 1)−1 + c0(K) +O(s− 1).In [Iha06 Y. Ihara made an extensive study of the Euler-Krone ker onstants of global elds,in parti ular, he obtained an asymptoti formula for their behaviour in families of urves overnite elds. A omplementary result in the number eld setting was obtain in [Zy10 as a orollary of theorem 3.9. In fa t the theorem 3.9 gives that in asymptoti ally exa t familiesthe oe ients of the Laurant series at s = 1 of the logarithmi derivatives ζ ′Ki

(s)/ζKi(s) tendto the orresponding oe ients of the Laurant series expansion of the logarithmi derivativeof the limit zeta fun tion. For zeroes oe ient this be omes:Corollary 3.12 (IharaZykin). Assuming GRH in the number eld ase and un ondi-tionally in the fun tion eld ase, for any asymptoti ally exa t family of global elds Ki wehave

limi→∞

γKi

gKi

= −∑

q

φqlog q

q − 1.For the sake of ompleteness let us mention an expli it analogue of theorem 3.9 obtained in[LZ:Theorem 3.13 (Leba queZykin). For any global eld K, any integer N ≥ 10 and any

ǫ = ǫ0 + iǫ1 su h that ǫ0 = Re ǫ > 0 we havePubli ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 59(i) in the fun tion eld ase:N

f=1

fΦrf

r(12+ǫ)f − 1

+1

log r· ZK

(

1

2+ ǫ

)

+1

r−12+ǫ − 1

= O

(

gK

rǫ0N

(

1 +1

ǫ0

))

+O(

rN2

)

;(ii) and in the number eld ase assuming GRH:∑

q≤N

Φq log q

q12+ǫ − 1

+ ZK

(

1

2+ ǫ

)

+1

ǫ− 12

=

= O

( |ǫ|4 + |ǫ|ǫ20

(gK + nK logN)log2N

N ǫ0

)

+O(√

N)

.3.4. Some other topi s related to limit zeta-fun tions. Let us nally state somerelated results on the asymptoti properties of the oe ients of zeta fun tions. For themoment they are only available in the fun tion eld ase (see [TV97). Let K/Fr(t) bea fun tion eld and let ζK(s) =∞∑

m=1Dmr

−ms be the Diri hlet series expansion of the zetafun tion of K. One knows that Dm is equal to the number of ee tive divisors of degree m onthe orresponding urve. We have the following results on the asymptoti behaviour of Dm :Theorem 3.14 (TsfasmanVl duµ). For an asymptoti ally exa t family of fun tionelds K = Ki and any real µ > 0 we havelimi→∞

logD[µg](Ki)

gKi

= mins≥1

(µs log q + log ζK(s)).Moreover, the minimum an be evaluated expli itly via φq ( .f. [TV97, Proposition 4.1).Theorem 3.15 (TsfasmanVl duµ). For an asymptoti ally exa t family of fun tionelds K = Ki, any ǫ > 0 and any m su h that Dm

g ≥ µ1 + ǫ we havelogDm(Ki)

hKi

=qm−g+1

q − 1(1 + o(1))for g → ∞, o(1) being uniform in m. Here µ1 is the largest of the two roots of the equation

µ

2+ µ logr

µ

2+ (2 − µ) logr

(

1 − µ

2

)

= −2 logr ζK(1).We should note that o(1) from theorem 3.15 is additive whereas most of the previous resultswere estimates of multipli ative type (they ontained logarithms of the quantities in question).It would be interesting to know whether there exist analogues of the above results in thenumber eld ase.Let us on lude by refering the reader to the Se tion 6 of [TV02 for a list of open questions.Publi ations mathématiques de Besançon - 2011

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60 Asymptoti methods in number theory and algebrai geometry4. Examples4.1. Towers of modular urves. Let us begin with the examples of asymptoti allyoptimal families of urves over nite elds oming from towers of modular urves. The rst onstru tions were arried out by Ihara ([Iha81), TsfasmanVl duµZink ([TVZ). Theresear h in this dire tion was ontinued by N. Elkies and many others. Let us des ribe several onstru tions.4.1.1. Classi al modular urves. Let us start with the onstru tion of towers of modular urves whi h leads to asymptoti ally optimal innite fun tion elds. For further information,we refer the reader to [TV92, Chapter 4. It is well known that the modular group Γ(1) =

PSL2(Z) a ts on the Poin aré upper half-plane h by (

a b

c d

)

· z =az + b

cz + d. We x a positiveinteger N and we dene the prin ipal ongruen e subgroup of level N by

Γ(N) =

γ ∈ Γ(1) | γ ≡(

1 0

0 1

)

mod N

.

Γ(N) ⊳ Γ(1) and Γ(1)/Γ(N) is isomorphi to PSL2(Z/NZ). In parti ular,[Γ(1) : Γ(N)] =

N3

2

ℓ|N

(

1 − ℓ−2) si N ≥ 3

6 si N = 2.We also put Γ0(N) =

γ ∈ Γ(1) | γ ≡(

∗ ∗0 ∗

)

mod N

, so that Γ(N) ⊂ Γ0(N). We have[Γ(1) : Γ0(N)] = N

ℓ|N

(

1 − ℓ−1)

.Let now Γ be a ongruen e subgroup, that is, any subgroup of Γ(1) ontaining Γ(N). The mostimportant ase for us is Γ = Γ(N) or Γ0(N). The set YΓ = Γ\h is equipped with an analyti stru ture, but is not ompa t. To ompa tify it we add points at innity (named usps):Γ(1) a ts naturally on P1(Q) and we put XΓ = (Γ\h) ∪ (Γ\P1(Q)). This way it be omes a onne ted Riemann surfa e alled modular urve. We let X(N) = XΓ(N), X0(N) = XΓ0(N),

Y (N) = YΓ(N) and Y0(N) = XΓ0(N).If Γ′ ⊂ Γ ⊂ Γ(1), there is a natural proje tion from XΓ′ → XΓ, whi h allows us to omputethe genus of the modular urve using the overing (the fun tion j is in fa t the j-invariant ofthe ellipti urve C/(Z + zZ)):XΓ

// XΓ(1)∼

j// P1(C)via the Hurwitz formula. For instan e,

gX(N) = 1 +(N − 6)[Γ(1) : Γ(N)]

12N.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 61It an be shown that Y (1) lassies isomorphism lasses of omplex ellipti urves and thatY0(N) lassies pairs (E,CN ), E being a omplex ellipti urve and CN being a y li sub-group of E of order N.Now, to onstru t towers of urves dened over nite elds, we need to take redu tions ofour modular urves modulo primes. If S is a s heme and E → S is an ellipti urve, the setof se tions E(S) is an abelian group. Let EN (S) denote the points of order dividing N inE(S). We all a level N stru ture an isomorphism αN : EN (S) → (Z/NZ)2. One an provethat there exists a smooth ane s heme Y (N) over SpecZ[1/N ] lassifying the isomorphism lasses of pairs (E,αN ) onsisting of an ellipti urve E/SpecZ[1/N ] together with a level Nstru ture αN on E. One an prove that this urve is a model of Y (N) over SpecZ[ζN , 1/N ],where ζN is a primitive N th-root of 1. There is also a model of Y0(N) over SpecZ[1/N ] andthis oarse moduli spa e lassies pairs onsisting of an ellipti urve together with a y li subgroup of order N. Models for X(N) and X0(N) an also be obtained in su h a way thatthey be ome ompatible with those for Y (N) and Y0(N). These urves have good redu tionover any prime ideal not dividing N. Moreover, the urve X0(N) an be dened over Q andhas good redu tion at any prime number not dividing N. Let p be su h prime. We denoteby C0,N the urve over Fp2 obtained by redu tion of X0(N) mod p. The urve X(N) anbe dened over the quadrati subeld of Q(ζN ) and has good redu tion at all the primesnot dividing N. Let CN be the redu tion of X(N) at a prime, i. e. a urve over Fp2. One an see that the genus of X0(N) and of X(N) is preserved under redu tion. The pointsof these urves orresponding to supersingular ellipti urves are Fp2-rational and there are[Γ(1) : Γ(N)]

12(p− 1) of them on CN . This leads to the following theorem:Theorem 4.1. (Ihara, TsfasmanVl duµZink) Let ℓ be a prime number not equal to p.The families Cℓn and C0,ℓn satisfy φp2 = p− 1 and therefore are asymptoti ally optimal.Note that the result for C0,ℓn an be dedu ed immediately from the orresponding result for

Cℓn .4.1.2. Shimura modular urves. Similar results on Shimura urves allow us to onstru tdire tly asymptoti ally optimal families over Fr with r = q2 = p2m, p prime. To do so,following Ihara, we start with a p-adi eld kp with N(p) = q = pm. Let Γ be a torsion-freedis rete subgroup of G = PSL2(R) × PSL2(kp) with ompa t quotient and dense proje tionto ea h of the two omponents of G (su h Γ's exist). Ihara proved the following results thatrelate the onstru tion of optimal urves to (anabelian) lass eld theory, and therefore areof great interest for us:Theorem 4.2. (Ihara [Iha08) To any subgroup Γ of G with the above properties one anasso iate a omplete smooth geometri ally irredu ible urve X over Fr of genus ≥ 2, togetherwith a set Σ onsisting of (q− 1)(g− 1) Fr-rational points of X su h that there is a anoni alisomorphism (up to onjuga y) from the pronite ompletion of Γ to Gal(KΣ/K) where KΣPubli ations mathématiques de Besançon - 2011

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62 Asymptoti methods in number theory and algebrai geometrydenotes the maximal unramied Galois extension of the fun tion eld K of X in whi h all thepla es orresponding to the points of Σ are ompletely split.An easy omputation leads to the following result:Corollary 4.3. For any square prime power r, there is a tower of urves dened over Frwith φr =√r − 1.In fa t, the ellipti modular urves X(N) that we onstru ted in the previous se tion orre-spond to Γ = PSL2(Z[1/p]) and its prin ipal ongruen e subgroups of level N.4.1.3. Drinfeld modular urves. The appli ability of Drinfeld modular urves to the prob-lem of onstru tion of optimal urves has been known sin e late 80's. The results we are goingto dis uss next an be found in [TV92.Let L be a eld of hara teristi p and let Lτ denote the ring of non- ommutative polyno-mials in τ, onsisting of expressions of the form n

i=0

aiτi, ai ∈ L, with multipli ation satisfying

τ · a = ap · τ for any a ∈ L. Let A = Fr[T ].A Drinfeld module is an Fr-homomorphism φ : A→ Lτ, a 7→ φa satisfying a few te hni al onditions. Let γ be the map γ : A → L sending a ∈ A to the term of φa of degree zero.Noti e that φ is determined by φT and γ by γ(T ). We onsider only Drinfeld modules of rank2 that is we assume that φT is a polynomial in τ of degree 2 and we put φT = γ(T )+gτ+∆τ2

(∆ 6= 0). More generally, one an dene Drinfeld modules over any A-s heme S.Just as in the lassi al ase, given a proper ideal I of A, one an dene a level I stru tureon φ. There is an ane s heme M(I) of nite type over A that parametrizes pairs (φ, λ),where φ is a Drinfeld module over S and λ is a level I stru ture. The s heme M(I) has a anoni al ompa ti ation: there exists a unique s heme M(I) ontaining M(I) as an opendense subs heme, whose bres over SpecA[I−1] are smooth omplete urves. The groupGL2(A/I) a ts naturally on M(I) by operating on the stru tures of level I and this a tionextends to M(I).From now on, let I be a prime ideal generated by a polynomial of degree m prime to q − 1.Now, onsider the smooth omplete (redu ible) urve X(I) = M(I) ⊗A Fq over Fq. Notethat the A-algebra stru ture on Fq is obtained through the redu tion mod T. Consider thesubgroup

Γ0(I) =

(

a b

c d

)

∈ GL2(A) | c ∈ I

and let Γ0(I) be the image of this subgroup in GL2(A/I). Finally, we onsider the smooth omplete absolutely irredu ible urve X0(I) = X(I)/Γ0(I). The image of M(I) −M(I) inX0(I) onsists of two Fq-rational points. Moreover, the following result holds.Theorem 4.4. The family X0(I), where I is a prime ideal of A generated by a poly-nomial of degree prime to q − 1, is an asymptoti ally exa t family of urves dened over Fq,satisfying φq2 = q − 1 and thus is optimal.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 63Moreover, N. Elkies proved in [Elk that the family of urves X0(Tn) whi h parametrizesnormalized Drinfeld modules (γ(T ) = 1,∆ = −1) with a level T n stru ture is asymptoti allyoptimal. He also related it to the expli it towers of Gar ia and Sti htenoth dis ussed in thenext subse tion.4.2. Expli it towers. In the last fteen years, Gar ia, Sti htenoth and many othersmanaged to onstru t asymptoti ally good towers expli itely in a re ursive way. Their in-terest omes from oding theory for su h towers provide asymptoti ally good odes via the onstru tion of Goppa. Let us give an example of su h expli it towers.Theorem 4.5. (Gar iaSti htenoth) Let r = q2 be a prime power. The tower Fn denedre ursively starting from the rational fun tion eld F0 = Fr(x0) using the relations Fn+1 =

Fn(xn+1), wherexq

n+1 + xn+1 =xq

n

xq−1n + 1

,satises φr =√r − 1 and thus is optimal.If the ardinality of the ground eld is not a square no towers with φr =

√r − 1 are known.However, there exist optimal towers in the sense that they have zero de ien y. Su h towers an be onstru ted starting from an expli it tower over a bigger eld using a des ent argument(see BalletRolland [BR for the details) or using modular towers.Let us now say a word about Elkies modularity onje ture. Elkies' work shows that most ofthe re ursive examples of Gar ia and Sti htenoth an be obtained by nding equations forsuitable modular towers. This made him formulate the following onje ture:Conje ture 4.6 (Elkies). Any asymptoti ally optimal tower is modular.Finally, let us note that there are other interesting onstru tions leading to expli it asymptot-i ally good towers of fun tion elds. As an example we mention the paper [BB by P. Beelenand I. Bouw who use Fu hsian dierential equations to produ e optimal towers over Fq2.4.3. Classeld towers. As it was said in se tion 2, tamely ramied innite extensions ofglobal elds with nitely many ramied pla es and with ompletely split pla es give examplesof asymptoti ally good towers. Given a global eld K, it is natural to onsider the maximalextension of K unramied outside a nite set of pla es S, in whi h pla es from a set T are ompletely split. But these extensions are very hard to understand. The maximal ℓ-extensionsare mu h easier to handle. These extensions are the limits of the ℓ-S-T - lass eld towers of

K.For a global eld K, two sets of nite pla es S and T (T 6= ∅(FF )) of K, and a prime numberℓ, onsider the maximal abelian ℓ-extension HT

S,ℓ(K) of K, unramied outside S and in whi hthe pla es from T are split (in the ase of fun tion elds the assumption on T to be non-emptyis made in order to avoid innite onstant eld extensions). Consider the tower re ursively onstru ted as follows: K0 = K, Ki+1 = HTS,ℓ(Ki). All the extensions Ki/K are Galois,Publi ations mathématiques de Besançon - 2011

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64 Asymptoti methods in number theory and algebrai geometryand we denote by GTS (K, ℓ) the Galois group Gal(

iKi,K). A su ient ondition for thistower to be innite is given by the GolodShafarevi h theorem: if G is a nite ℓ-group then

dimFℓH2(G,Fℓ) >

14 dimFℓ

H1(G,Fℓ)2. This allows to onstru t asymptoti ally good inniteglobal elds. The following result is at the base of many onstru tions of lass eld towerswith pres ribed properties:Theorem 4.7. [TsfasmanVl duµ [TV02 (NF), Serre [Ser85 , NiederreiterXing [NX(FF) Let K/k be a y li extension of global elds of degree ℓ. Let T (k) be a nite set of nonar himedean pla es of k and let T (K) be the set of pla es above T (k) in K. Suppose in thefun tion eld ase that GCDℓ,deg p, p ∈ T (K) = 1. Let Q be the rami ation lo us of K/k.Let

(FF ) C(T,K/k) =#T (k) + 2 + δℓ + 2√

#T (K) + δℓ,

(NF ) C(T,K/k) =#T (K) − t0 + r1 + r2 + δℓ + 2 − ρ+

2√

#T (K) + ℓ(r1 + r2 − ρ/2) + δℓ,where δℓ = 1 if K ontains the ℓ-root of unity, and 0 otherwise, t0 is the number of prin ipalideals in T (k), r1 = ΦR(K), r2 = ΦC(K) and ρ is the number of real pla es of k whi hbe ome omplex in K. Suppose that #Q ≥ C(T,K/k). Then K admits an innite unramedℓ-T (K)- lass eld tower.One an onstru t su h y li extension using the Grunwald-Wang theorem (and sometimeseven expli itly by hand) and dedu e the following result:Corollary 4.8 (Leba que). Let n be an integer and let t1, ..., tn be prime powers (NF)(powers of p (FF)). There exists an innite global eld (both in the number eld and fun tioneld ases) su h that φt1 , ..., φtn are all > 0.Another way to produ e asymptoti ally good innite lass eld towers is to use tamely ramiedinstead of unramied lass eld towers. This is the subje t of [HM01 and [HM02.The question of nding asymptoti ally good towers with given TsfasmanVl duµ invariantsequal to zero is more di ult. A related question is to nd out whether an innite globalextension realizes the maximal lo al extension at a given prime. Using results of J. Labute[Lab and A. S hmidt [S h, the following theorem is proven:Theorem 4.9 (Leba que [Leb09). Let P = p1, . . . , pn ⊂ Plf (Q). Assume that forany i = 1, . . . , n we have ni distin t positive integers di,1, . . . , di,ni

. Let I ⊂ Plf (Q) be a niteset of nite pla es of Q su h that I ∩ P = ∅. There exists an innite global eld K su h that:1. I ∩ Supp(K) = ∅,2. For any i = 1, . . . , n, and any j = 1, . . . , ni, φpi,Np

di,ji

= φ∞

nidi,j> 0.3. One an expli itly estimate φ∞ and the de ien y in terms of P, I, ni and dij .The φp,q are invariants generalizing the lassi al φq : they ount the asymptoti number ofprimes of norm q above a given prime p (see [Leb10 for a denition). In the ase of Q theyPubli ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 65 oin ide with the lassi al ones. This extension is obtained as the ompositum of a niteextension of Q with pres ribed positive Φpi,Npi

di,j > 0 and an innite lass eld tower QPS (ℓ)satisfying the K(π, 1) property of A. S hmidt.4.4. Bounds on the de ien y. We have already seen that, using towers of modular urves, one an produ e innite fun tion elds over Fr with zero de ien y. If r is a square,there are even towers with φr =

√r−1. In the ase of number elds no zero de ien y innitenumber elds are known. In fa t we doubt that the lass eld theory (whi h is for now theonly method to produ e asymptoti ally good innite number elds) an ever give su h eld.Let us quote here the example with the smallest known de ien y due to F. Hajir and Ch.Maire [HM02.Let k = Q(ξ), where ξ is a root of f(x) = x6 + x4 − 4x3 − 7x2 − x+ 1. Consider the element

η = −671ξ5 + 467ξ4 − 994ξ3 + 3360ξ2 + 2314ξ − 961 ∈ Ok. Let K = k(√η). F. Hajir and Ch.Maire prove using a GolodShafarevi h like result that K admits an innite tamely ramiedtower satisfying δ ≤ 0.137 . . . .5. Higher dimensional theoryIn this se tion we will mostly onsider the fun tion eld ase sin e most of the results weare going to mention are unavailable in the number eld ase. However, we will give somereferen es to the number eld ase as well.5.1. Number of points on higher dimensional varieties. The question about themaximal number of points on urves over nite elds has been extensively studied by numerousauthors. The analogous question for higher dimensional varieties has re eived omparativelylittle attention most probably due to its being signi antly more di ult.As for the urves, we have the so- alled Weil bound whi h is in this ase a famous theoremof Deligne. Similarly, this bound is not optimal and the general framework for improving itis provided by the expli it formulae. In the ase of urves over Fr Oesterlé managed to ndthe best bounds available through the te hniques of expli it formulae for any given r 6= 2 (see[Ser85). A de ade later the ase of arbitrary varieties over nite elds was treated by G.La haud and M. A. Tsfasman in [Tsf95 and [LT. Let us reprodu e here the main resultsfrom [LT. To do so we will have to introdu e some notation on erning varieties over niteelds.Let X be a non-singular absolutely irredu ible proje tive variety of dimension d dened overa nite eld Fr. We put Xf = X ⊗Fr Fqr and X = X ⊗Fr Fr. Let Φrf = Φrf (X) be thenumber of points of X having degree f . Thus, for the number Nf of Frf -points of the variety

Xf we have the formula Nf =∑

m|f

mΦrm. We denote by bs(X) = dimQlHs(X,Ql) the l-adi Betti numbers of X.The family of inequalities proven in [LT has a doubly positive sequen e as a parameter. Letus introdu e the orresponding notation. To a sequen e of real numbers v = (vn)n≥0 wePubli ations mathématiques de Besançon - 2011

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66 Asymptoti methods in number theory and algebrai geometryasso iate the family of power series ψm,v(t) =∞∑

n=1vmnt

n. We denote ψv(t) = ψ1,v(t) andlet ρv be the radius of onvergen e of this power series. A doubly positive sequen e v issu h a sequen e that 0 ≤ vn ≤ v0 for all n, v0 = 1 and for any z ∈ C, |z| < 1 we have1 + 2Reψv(t) ≥ 0.We will also need the fun tions Fm,v(k, t) =

∞∑

s=0(−1)sψm,v(r−kst) =

∞∑

n=1

vmntmn

1+r−mnk , Fv(k, t) =

F1,v(k, t). We let Av(z) = −min|t|=z

Reψv(t) and denote I(k) = i | 1 ≤ i ≤ 2d − 1, i 6= k, i 6=2d− k the set of indi es. We have the following inequalities:Theorem 5.1 (La haudTsfasman). For any odd integer k, 1 ≤ k ≤ d, any doublypositive sequen e v = (vn)n≥0 with ρv > qk/2 and any M ≥ 1 we have

M∑

m=1

mΦrm(X)ψm,v(r−(2d−k)/2) ≤ ψv(r−(2d−k)/2) + ψv(rk/2) +bk2

+

+∑

i odd,i6=k

biAv(r−(i−k)/2) +∑

i even biψv(r−(i−k)/2),andM∑

m=1

mΦrm(X)Fm,v(d− k, r−(2d−k)/2) ≤ Fv(d− k, r−(2d−k)/2) + Fv(d− k, rk/2)+

+bk2

+∑

i∈I(k)

biFv(d− k, r−(i−k)/2).For example, taking the se ond inequality with ψv(t) = t2 we get the lassi al Weil bound,taking the rst one with ψv(t) = t

1−t we get (asymptoti ally) a dire t generalization of theDrinfeldVl duµ bounds. These inequalities are not straightforward to apply. We refer thereader to [LT for more details on how to make good hoi es of the doubly positive sequen e.Unfortunately, in the ase of dimension d ≥ 2 the optimal hoi e of v is unknown.The asymptoti versions of these inequalities an be easily dedu ed from theorem 5.1 on eone introdu es proper denitions. For a variety X let b(X) = maxi=0,...,d

bi(X) be the maximumof its l-adi Betti numbers.Denition 5.2. A family of varieties Xj is alled asymptoti ally exa t if the limitsφrf = lim

j→∞

Φrf (Xj)

b(Xj ) and βi = limj→∞

bi(Xj)b(Xj ) exist. It is asymptoti ally good if at least one of φrfis dierent from zero.We an state the following orollary of theorem 5.1:Corollary 5.3. In the notation of theorem 5.1 for an asymptoti ally exa t family of vari-eties one hasPubli ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 67M∑

m=1

mφrmψm,v(r−(2d−k)/2) ≤ βk

2+

i odd,i6=k

βiAv(r−(i−k)/2) +∑

i evenβiψv(r−(i−k)/2),andM∑

m=1

mφrmFm,v(d− k, r−(2d−k)/2) ≤ βk

2+

i∈I(k)

βiFv(d− k, r−(i−k)/2).Taking parti ular examples of the sequen e v one gets more tra table inequalities (see [LT).5.2. BrauerSiegel type onje tures for abelian varieties over nite elds. One an ask about the possibility of extending the BrauerSiegel theorem to the ase of varietiesover nite elds. The question is not as easy as it might seem. First, mimi king the proof oftheorem 3.4 one gets a result about the asymptoti behaviour of the residues of zeta fun tionsof varieties at s = d (see [Zyk09). Su h a result would be interesting if there was a reasonableinterpretation for this residue in terms of geometri invariants of our variety.Two other approa hes were suggested by B. Kunyavskii and M. Tsfasman and by M. Hindryand A. Pa he o. Both of them have for their starting points the Bir h and Swinnerton-Dyer(BSD) onje ture whi h expresses the value at s = 1 of the L-fun tion of an abelian variety interms of ertain arithmeti invariants related to this variety. However, the situation with theasymptoti behaviour of this spe ial value of the L-fun tions is mu h less lear than before.Let us begin with the approa h of Kunyavskii and Tsfasman.Let K/Fr be a fun tion eld and let A/K be an abelian variety over K. We denote byXA := |X(A/K)| the order of the Shafarevi hTate group of A, and by RegA the determinantof the MordellWeil latti e of A (see [HP for denitions). Note that in a ertain sense XAand RegA are the analogues of the lass number and of the regulator respe tively. Kunyavskiiand Tsfasman make the following onje ture on erning families of onstant abelian varieties(see [KT):Conje ture 5.4. Let A0 be a xed abelian variety over Fr. Take an asymptoti ally exa tfamily of fun tion eds K = Ki and put Ai = A0 ×Fr Ki. Then

limi→∞

logr(Xi · Regi)

gi= 1 −

∞∑

m=1

φrm(K) logr

|A0(Frm)|rm

.This onje ture is a tually stated as theorem in [KT. Unfortunately the hange of limits inthe proof given in [KT is not justied thus the proof an not be onsidered a valid one. Infa t the aw looks very di ult to repair as the statement of the theorem an be redu ed(via a formula due to J. Milne, whi h gives the BSD onje ture in this ase) to an equalityof the type limi→∞

log ζKi(s)

gKi

= log ζK(s) at a given point s ∈ C with Re s = 12 (in fa t s belongsto a nite set of points depending on A0). As we have already mentioned in the dis ussionfollowing theorem 3.10 this question does not look a essible at the moment.Publi ations mathématiques de Besançon - 2011

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68 Asymptoti methods in number theory and algebrai geometryLet us turn our attention to the approa h of Hindry and Pa he o. They treat the ase in somesense orthogonal to that of Kunyavskii and Tsfasman. Here is the onje ture they make in[HP:Conje ture 5.5. Consider the family Ai of non- onstant abelian varieties of xed di-mension over the xed fun tion eld K. We havelimi→∞

log(Xi · Regi)

logH(Ai)= 1,where H(Ai) is the exponential height of Ai.Using deep arguments from the theory of abelian varieties over fun tion elds the onje ture isredu ed in [HP to the one on zeroes of L-fun tions of abelian varieties together with the BSD onje ture. Hindry and Pa he o are a tually fa ed with the problem of the type dis ussedafter theorem 3.10, this time for abelian varieties over fun tion elds.The following example serves as the eviden e for the last onje ture (see [HP):Theorem 5.6 (HindryPa he o). For the family of ellipti urves Ed over Fr(t), wherethe hara teristi of Fr is not equal to 2 or 3, dened by the equations y2 +xy = x3− td, d ≥ 1and prime to r, the TateShafarevi h group X(Ed/K) is nite and

log(Xd · Regd) ∼ logH(Ed) ∼d log r

6.The proof of this theorem uses a deep result of Ulmer [Ulm02 who established the BSD on-je ture in this ase and expli itly omputed the L-fun tions of Ed. This redu es the statementof the theorem to a an expli it (though highly non-trivial) estimate involving Ja obi sums.The onje tures 5.4 and 5.5 an be united (though not proved) within the general asymptoti theory of L-fun tions over fun tion elds. Su h a theory also explains why we get 1 as a limitin the se ond onje ture and a ompli ated expression in the rst one. We will sket h someaspe ts of the theory in the next subse tion.The analogous problem in the number eld ase has also been onsidered [Hin. Unfortunatelyin the number eld ase we do not have a single example supporting the onje ture.5.3. Asymptoti theory of zeta and L-fun tions over nite elds. The proofs ofthe results from this subse tion as well as lengthy dis ussions an be found in [Zyk11. Letus rst dene axiomati ally the lass of fun tions we are going to work with. This resemblesthe so alled Selberg lass from the analyti number theory, but, of ourse the ase of fun tionelds is innitely easier from the analyti point of view, all fun tions being rational (or evenpolynomial).Denition 5.7. An L-fun tion L(s) over a nite eld Fr is a holomorphi fun tion in ssu h that for u = q−s the fun tion L(u) = L(s) is a polynomial with real oe ients, L(0) = 1and all the roots of L(u) are on the ir le of radius r− d

2 for some non-negative integer numberd whi h is alled the weight of the L-fun tion. We say that the degree of the polynomial L(u)Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 69is the degree of the orresponding L-fun tion. A zeta fun tion ζ(s) is a produ t of L-fun tionsin powers ±1 :

ζ(s) =d

k=0

Lk(s)wk ,where wk ∈ −1, 1 and Lk(s) is an L-fun tion of weight k.Both zeta-fun tions of smooth proje tive urves or even varieties over nite elds and L-fun tions of ellipti surfa es onsidered in the previous se tions are overed by this denition.For the logarithm of a zeta fun tion we have the Diri hlet series expansion:

log ζ(s) =

∞∑

f=1

Λf

fr−fswhi h is onvergent for Re s > d

2 . In the ase of a variety X/Fr we have a simple interpretationfor the oe ients Λf = |X(Frf )| as the number of points on X over the degree f extensionof Fr.We are going to work with zeta and L-fun tions asymptoti ally, so we have to introdu e thenotion of a family. We will all a sequen e ζk(s)k=1...∞ =

d∏

i=0Lki(s)

wi

k=1...∞

of zetafun tions a family if the total degree gk =d∑

i=0gki tends to innity and d remains onstant.Here gki are the degrees of the individual L-fun tions Lki(s) in ζk(s).Denition 5.8. A family ζk(s)k=1...∞ of zeta-fun tions is alled asymptoti ally exa tif the limits

γi = limk→∞

gki

gkand λf = lim

k→∞

Λkf

gkexist for ea h i = 0, . . . , d and ea h f ∈ Z, f ≥ 1. The family is alled asymptoti ally bad ifλf = 0 for any f and asymptoti ally good otherwise.In the ase of urves over nite elds the denominators of zeta fun tions are negligible fromthe asymptoti point of view. In general we give the following denition:Denition 5.9. Let ζk(s) be an asymptoti ally exa t family of zeta fun tions. Denethe set I ⊂ 0 . . . d by the ondition i ∈ I if and only if γi = 0.We dene ζn,k(s) =

i∈I

Lki(s)withe negligible part of ζk(s) and ζe,k(s) =

i∈0,...,d\I

Lki(s)wi the essential part of ζk(s). Denealso de = maxi | i /∈ I.Denition 5.10. We say that an asymptoti ally exa t family of zeta or L-fun tions isasymptoti ally very exa t if the series

∞∑

f=1

|λf |q−fde2 Publi ations mathématiques de Besançon - 2011

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70 Asymptoti methods in number theory and algebrai geometryis onvergent.In the ase of urves or varieties the positivity of Λf automati ally implies the fa t thatthe orresponding family is asymptoti ally very exa t. This is of ourse false in general (anobvious example of a family whi h is asymptoti ally exa t but not very exa t is given byLk(s) = (1 − q−s)k). In general most of the results are proven for asymptoti ally very exa tfamilies and not just for asymptoti ally exa t ones.We have already noted that the on ept of limit zeta fun tions is of utmost importan e in theasymptoti theory.Denition 5.11. Let ζk(s) be an asymptoti ally exa t family of zeta fun tions. Thenthe orresponding limit zeta fun tion is dened as

ζlim(s) = exp

∞∑

f=1

λf

fq−fs

.Now, we an state the generalizations of most of the results on erning zeta and L-fun tionsover nite elds, given in the previous se tions. Let us begin with the basi inequalities. Infa t, one should be able to write most of the inequalities from subse tion 5.1 in this moregeneral setting. We give only the simplest statement of this type here:Theorem 5.12. Let ζk(s) be an asymptoti ally very exa t family of zeta fun tions. Thenwde

∞∑

j=1

λjq− dej

2 ≤de∑

i=0

γi

q(de−i)/2 + wi.The BrauerSiegel type results an also be proven in this setting. The following theoremin ludes all the fun tion eld versions of the BrauerSiegel type results from se tion 3 ex eptfor the expli it ones (whi h an also be, in prin iple, established for general zeta and L-fun tions).Theorem 5.13. 1. For any asymptoti ally exa t family of zeta fun tions ζk(s) andany s with Re s > de

2 we havelim

k→∞

log ζe,k(s)

gk= log ζlim(s).If, moreover, 2Re s 6∈ Z, then

limk→∞

log ζk(s)

gk= log ζlim(s).The onvergen e is uniform in any domain de

2 + ǫ < Re s < de+12 − ǫ, ǫ ∈

(

0, 12

)

.2. If ζk(s) is an asymptoti ally very exa t family with wde= 1 we have:

limk→∞

log |ck|gk

≤ log ζlim

(

de2

)

,Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 71where rk and ck are dened using the Taylor series expansion ζk(s) = ck(

s− de

2

)rk +

O(

(

s− de

2

)rk+1)

.In the ase of arbitrary L-fun tions the equality in (2) does not hold in general. This meansthat the similar questions previously dis ussed for fun tion elds or ellipti urves over fun -tion elds are indeed of arithmeti nature.Finally we will state a result on the distribution of zeroes. Let L(s) be an L-fun tion andlet ρ1, . . . , ρg be the zeroes of the orresponding polynomial L(u). Dene θk ∈ (−π, π] byρk = q−d/2eiθk . One an asso iate the measure ∆L = 1

g

g∑

k=1

δθkto L(s).Theorem 5.14. Let Lj(s) be an asymptoti ally very exa t family of L-fun tions. Thenthe limit distribution lim

j→∞∆j exists and has a nonnegative ontinuous density fun tion givenby an absolutely and uniformly onvergent series 1 − 2

∞∑

k=1

λk cos(kx)q−dk2 .In the ase of families of ellipti urves over Fr(t) P. Mi hel provides in [Mi an expli itestimate for the dis repan y in the equidistribution of zeroes and a mu h more pre ise estimatefor it on average.A number of open questions on erning asymptoti properties of zeta and L-fun tions an befound in the last se tion of [Zyk11. It seems that an analogue of this general asymptoti theory an be developed in the number eld ase (at least assuming some plausible onje tureslike GRH or the RamanujanPeterson onje ture). This is yet to be done.Referen es[BB P. Beelen, I. Bouw. Asymptoti ally good towers and dierential equations, Compos. Math. 141(2005), no. 6, 1405-1424.[BR S. Ballet, R. Rolland. Families of urves over nite elds, PMB 2011, 5-18.[Bra R. Brauer. On zeta-fun tions of algebrai number elds, Amer. J. Math. 69, Num. 2, 1947,243250.[DPZ I. Duursma, B. Poonen, M. Zieve. Everywhere ramied towers of global fun tion elds, Finiteelds and appli ations, Le ture Notes in Comput. S i., vol. 2948, Springer, Berlin, 2004, pp. 148-153.[DV V. G. Drinfeld, S. G. Vl duµ. The number of points of an algebrai urve (in Russian), Funkt-sional. Anal. i Prilozhen. 17 (1983), no. 1, 6869.[Elk N. D. Elkies. Expli it towers of Drinfeld modular urves, Progress in Mathemati s 202 (2001),189-198 (Pro eedings of the 3rd European Congress of Mathemati s, Bar elona.[GSR A. Gar ia, H. Sti htenoth, H.-G. Rü k. On tame towers over nite elds, Journal für die Reineund Angewandte Mathematik, vol. 557(2003).[Hin M. Hindry. Why is it di ult to ompute the MordellWeil group. Pro eedings of the onferen eDiophantine Geometry, 197219, Ed. S uola Normale Superiore Pisa, 2007.[HM01 F. Hajir, C. Maire. Tamely ramied towers and dis riminant bounds for number elds. Com-positio Math. 128 (2001), no. 1, 35-53.[HM02 F. Hajir, C. Maire. Tamely ramied towers and dis riminant bounds for number elds. II. J.Symboli Comput. 33 (2002), no. 4, 415-423. Publi ations mathématiques de Besançon - 2011

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72 Asymptoti methods in number theory and algebrai geometry[HP M. Hindry, A. Pa he o. An analogue of the Brauer-Siegel theorem for abelian varieties in positive hara teristi , preprint.[Iha81 Y. Ihara. Some remarks on the number of rational points of algebrai urves over nite elds,J. Fa . S i. Univ. Tokyo Se t. IA Math., 28(3):721724 (1982), 1981.[Iha83 Y. Ihara. How many primes de ompose ompletely in an innite unramied Galois extensionof a global eld? J. Math. So . Japan 35(1983), n.4, 693709.[Iha06 Y. Ihara. On the EulerKrone ker onstants of global elds and primes with small norms,Algebrai geometry and number theory, 407451, Progr. Math., 253, Birkhaüser Boston, Boston,MA, 2006.[Iha08 Y. Ihara. On ongruen e monodromy problems, MSJ Memoirs 18, Math. So . Japan, 2008.[IS99 H. Iwanie , P. Sarnak. Diri hlet L-fun tions at the entral point. Number theory in progress,Vol. 2 (Zakopane-Kos ielisko, 1997), 941952, de Gruyter, Berlin, 1999.[IS00 H. Iwanie , P. Sarnak. The nonvanishing of entral values of automorphi L-fun tions andSiegel's zero, Israel J. Math. 120 (2000), 155-177.[KT B. E. Kunyavskii, M. A. Tsfasman. BrauerSiegel theorem for ellipti surfa es, Int. Math. Res.Not. IMRN 2008, no. 8.[Lab J. Labute. Mild pro-p-groups and Galois groups of p-extensions of Q, J. Reine Angew. Math.,596:155182, 2006.[Lan71 S. Lang. On the zeta fun tion of number elds, Invent. Math. 12 (1971), 337345.[Lan94 Lang, S. Algebrai number theory (Se ond Edition), Graduate Texts in Mathemati s 110,Springer-Verlag, New York, 1994.[LT G. La haud, M. A. Tsfasman. Formules expli ites pour le nombre de points des variétés sur un orps ni, J. Reine Angew. Math. 493 (1997), 160.[Leb07 P. Leba que. Generalised Mertens and BrauerSiegel Theorems, A ta Arith. 130 (2007), no.4, 333350.[Leb09 P. Leba que. Quelques résultats ee tifs on ernant les invariants de Tsfasman-Vl duµ,preprint arXiv:0903.3027.[Leb10 P. Leba que. On TsfasmanVl duµ invariants of innite global elds, Int. J. Number Theory,no. 6(2010), 14191448.[LZ P. Leba que, A. Zykin. On logarithmi derivatives of zeta fun tions in families of global elds,to appear at Int. J. Number Theory.[Mi P. Mi hel. Sur les zéros de fon tions L sur les orps de fon tions, Math. Ann. 313 (1999), no.2, 359370.[NX H. Niederreiter, C. Xing. Rational points on urves over nite elds: theory and appli ations.London Mathemati al So iety Le ture Note Series, 285. Cambridge University Press, Cambridge,2001.[Odl76 A. M. Odlyzko. Lower bounds for dis riminants of number elds. A ta Arith. 29(1976), 275297.[Odl90 A. M. Odlyzko. Bounds for dis riminants and related estimates for lass numbers, regulatorsand zeroes of zeta-fun tions: a survey of re ent results. Sém. Th. Nombres Bordeaux, 1990, v.2,119141.[S h A. S hmidt. Über pro-p-fundamentalgruppen markierter arithmetis her Kurven, J. ReineAngew. Math., 640:203235, 2010.[Ser75 J.-P. Serre. Minorations de dis riminants, note of O tober 1975, published on pp. 240-243 invol. 3 of Jean-Pierre Serre, Colle ted Papers, Springer 1986.[Ser85 J.-P. Serre. Rational points on urves over Finite Fields, Notes of Le tures at Harvard Uni-versity by F. Q. Gouvêa, 1985.Publi ations mathématiques de Besançon - 2011

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Philippe Leba que and Alexei Zykin 73[Sie C. L. Siegel. Über die Classenzahl quadratis her Zahlkörper. A ta Arith. 1 (1935), 8386.[Sta74 H. M. Stark. Some ee tive ases of the BrauerSiegel Theorem, Invent. Math. 23(1974),135152.[Tsf92 M. A. Tsfasman. Some remarks on the asymptoti number of points, Coding Theory andAlgebrai Geometry, Le ture Notes in Math. 1518, 178192, SpringerVerlag, Berlin 1992.[Tsf95 M. A. Tsfasman. Nombre de points des surfa es sur un orps ni, Algebrai Geometry andCoding Theory, Pro eedings AGCT-4, De Gruyter, 1995.[TV92 M. A. Tsfasman, S. G. Vl duµ. Algebrai -geometri odes. Translated from the Russian by theauthors. Mathemati s and its Appli ations (Soviet Series), 58. Kluwer A ademi Publishers Group,Dordre ht, 1991.[TV97 M. A. Tsfasman, S. G. Vl duµ. Asymptoti properties of zeta-fun tions, J. Math. S i. 84(1997), Num. 5, 14451467.[TV02 M. A. Tsfasman, S. G. Vl duµ. Ininite global elds and the generalized BrauerSiegel The-orem, Mos ow Mathemati al Journal, Vol. 2 (2002), Num. 2, 329402.[TVN M. A. Tsfasman, S. G. Vl duµ, D. Nogin. Algebrai geometri odes: basi notions, Mathe-mati al Surveys and Monographs, 139, Ameri an Mathemati al So iety, Providen e, RI, 2007.[TVZ M. A. Tsfasman, S. G. Vl duµ, Th. Zink. Modular urves, Shimura urves, and Goppa odes,better than Varshamov-Gilbert bound, Math. Na hr. 109(1982), 2128.[Ulm02 D. Ulmer. Ellipti urves with high rank over fun tion elds, Annals of Math. 155(2002),295315.[Zim R. Zimmert. Ideale kleiner Norm in Idealklassen und eine Regulatorabs hatzung, Invent. Math.62(1981), 367380.[Zyk05 A. Zykin. BrauerSiegel and TsfasmanVl duµ theorems for almost normal extensions ofglobal elds, Mos ow Mathemati al Journal, Vol. 5 (2005), Num 4, 961968.[Zyk09 A. Zykin. On the generalizations of the BrauerSiegel theorem. Pro eedings of the Conferen eAGCT 11 (2007), Contemp. Math. series, AMS, 2009.[Zy10 A. Zykin. Asymptoti properties of Dedekind zeta fun tions in families of number elds, Jour-nal de Théorie des Nombres de Bordeaux 22(2010), no. 3, 689696.[Zyk11 A. Zykin. Asymptoti properties of zeta fun tions over nite elds, preprint.02 mai 2011Philippe Leba que, Laboratoire de Mathématiques de Besançon, UFR S ien es et te hniques, 16, route deGray 25 030 Besançon, Fran e • E-mail : philippe.leba queuniv-f omte.frAlexey Zykin, State University Higher S hool of E onomi s, Department of Mathemati s, Laboratory ofAlgebrai Geometry, SU-HSE, 117312, 7 Vavilova Str., Mos ow, RussiaInstitute for Information Transmission Problems of the Russian A ademy of S ien es, LaboratoirePon elet (UMI 2615) • E-mail : alzykingmail. omPubli ations mathématiques de Besançon - 2011

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A COMPUTATIONAL STUDY OF THE ASYMPTOTIC BEHAVIOUROF COEFFICIENT FIELDS OF MODULAR FORMSbyMar el Mohyla & Gabor WieseAbstra t. The arti le motivates, presents and des ribes large omputer al ulations on- erning the asymptoti behaviour of arithmeti properties of oe ient elds of modular forms.The observations suggest ertain patterns, whi h deserve further study.Résumé. Le but de et arti le est de motiver, présenter et dé rire de nombreux al ulsmenés sur ordinateur on ernant le omportement asymptotique de propriétés arithmétiquesdes orps des oe ients de formes modulaires. Les observations suggèrent plusieurs questionsqui méritent d'être étudiées ultérieurement.1. Introdu tionA re ent breakthrough in Arithmeti Geometry is the proof of the Sato-Tate onje ture byBarnet-Lamb, Clozel, Geraghty, Harris, Shepherd-Barron and Taylor ([BLGHT, [CHT,[HSHT, [T). It states that the normalised He ke eigenvalues ap(f)

2p(k−1)/2 on a holomorphi newform f of weight k ≥ 2 (and trivial Diri hlet hara ter(1)) are equidistributed with respe tto a ertain measure (the so- alled Sato-Tate measure), when p runs through the set of primenumbers. The name horizontal Sato-Tate is sometimes used for this situation.The reversed situation, to be referred to as verti al horizontal Sato-Tate, was su essfullytreated by Serre in [S. He xes a prime p and allows any sequen e of positive integers(Nn, kn) with even kn and p ∤ Nn su h that Nn +kn tends to innity and proves that ap(f)

2p(k−1)/22000 Mathemati s Subje t Classi ation. 11F33 (primary); 11F11, 11Y40.Key words and phrases. Modular forms, oe ient elds, asymptoti behaviour, ongruen es, He kealgebras.G. W. would like to thank Gabriel Chênevert for inspiring dis ussions of variants of Sato-Tate. More thanks aredue to Pierre Parent, Christophe Ritzenthaler, René S hoof and Mark Watkins for interesting dis ussions andsuggestions. G. W. a knowledges partial support by the European Resear h Training Network Galois Theoryand Expli it Methods MRTN-CT-2006-035495 and by the Sonderfors hungsberei h Transregio 45 Periods,moduli spa es and arithmeti of algebrai varieties of the Deuts he Fors hungsgemeins haft.(1)We only make this assumption for the sake of simpli ity of the exposition.

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76 Asymptoti behaviour of oe ient elds of modular formsis equidistributed with respe t to a ertain measure depending on p (whi h is related to theSato-Tate measure), when f runs through all the newforms in any of the spa es of usp formsof level Γ0(Nn) and weight kn. A orollary is that for xed positive even weight k, the set(1.1) [Qf : Q] | f newform of level Γ0(Nn) and weight kis unbounded for any sequen e Nn tending to innity. Here, Qf denotes the number eldobtained from Q by adjoining all He ke eigenvalues on f .In this arti le we perform a rst omputational study towards a (weak) arithmeti analogue ofverti al Sato-Tate, where the name arithmeti refers to taking a nite pla e of Q, as opposed tothe innite pla e used in usual Sato-Tate (the assertion of usual Sato-Tate on erns the He keeigenvalues as real numbers). An arithmeti analog of horizontal Sato-Tate is Chebotarev'sdensity theorem. Consider, for example, a normalised uspidal He ke eigenform f with at-ta hed Galois representation ρf : Gal(Q/Q) → GL2(Zℓ).(2) Fix some x ∈ Zℓ and let n ∈ N.Let G be the image of the omposite representation Gal(Q/Q)ρf−→ GL2(Zℓ) ։ GL2(Z/ℓnZ)and let d(x) be the number of elements in G with tra e equal to x modulo ℓn. Then the den-sity of the set p | |ap − x|ℓ ≤ ℓ−n is equal to d(x)

|G| by Chebotarev's density theorem; hen e,the situation is ompletely lear. Whereas at the innite pla e horizontal Sato-Tate seems tobe more di ult than verti al Sato-Tate, the situation appears to be reversed for arithmeti analogs. We are not going to propose su h an analogue. But, we are going to study relatedquestions by means of omputer al ulations. For instan e, as a motivation let us onsiderthe set(1.2) [Fℓ,f : Fℓ] | f ∈ Sk(Nn; Fℓ) normalised He ke eigenformin analogy to Equation 1.1. Here, Sk(Nn; Fℓ) denotes the Fℓ-ve tor spa e of uspidal modularforms over Fℓ (see Se tion 2 for denitions) and Fℓ,f is dened by adjoining to Fℓ all He keeigenvalues on f . It is easy to onstru t sequen es (Nn, kn) for whi h the set in question isinnite (see e.g. [DiWi and [W), but it does not seem simple to obtain all natural numbersas degrees. Most importantly, it seems to be unknown whether this set is innite when (Nn)is the sequen e of prime numbers, k = 2 and ℓ > 2.Con erning properties of modular forms in positive hara teristi , there is other, mu h moresubtle information than just the degrees of Fℓ,f to be studied, e.g. ongruen es between mod-ular forms. In order to take the full information into a ount, in this arti le we examine theproperties of the Fℓ-He ke algebras Tk(Nn) on Sk(Nn; Fℓ) asymptoti ally for xed weight k(mostly 2) and running level Nn (mostly the set of prime numbers) by means of experimen-tation. More pre isely, we investigate three quantities:(a) The deviation of Tk(Nn) from being semisimple. In Se tion 3, we in lude a propositionrelating nonsemisimpli ity to ongruen es, rami ation and ertain indi es. Our exper-iments suggest that for odd primes ℓ, the He ke algebra Tk(Nn) tends to be lose to(2)Again, it is only for simpli ity of the exposition that we are taking Zℓ instead of Qℓ.Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 77semisimple, whereas the situation seems to be ompletely dierent for p = 2 (see Se -tion 4.1).(b) The average residue degree of Tk(Nn). That is the arithmeti mean of the degrees ofFℓ,f for all f in Sk(Nn; Fℓ). Our omputations (see Se tion 4.2) strongly suggest thatthis quantity is unbounded. More pre isely, we seem to observe a ertain asymptoti behaviour, whi h we formulate as a question.( ) The maximum residue degree of Tk(Nn). That is the maximum of the degrees of Fℓ,ffor all f in Sk(Nn; Fℓ). Our experiments suggest that this quantity is 'asymptoti ally'proportional to the dimension of Sk(Nn; Fℓ).In Se tion 4 we des ribe our omputations and derive ertain questions from our observations.However, we do not attempt to propose any heuristi explanations in this arti le. This willhave to be the subje t of subsequent studies, building on rened and extended omputations.We see (b) and ( ) as strong eviden e for the innity of the set in Equation 1.2, when Nnruns through the primes. Generally speaking, there appears to be some regularity in theotherwise quite errati behaviour of the examined quantities, lending some support to thehope of nding a formulation of an arithmeti analogue of verti al Sato-Tate.2. Ba kground and notationWe start by introdu ing the ne essary notation and explaining the ba kground. For fa tson modular forms, we refer to [DI. Let us x an interger k and a ongruen e subgroup

Γ ⊆ SL2(Z). Denote by Sk(Γ) the omplex ve tor spa e of holomorphi usp forms of weight kfor Γ. Dene T := Tk(Γ) to be the Z-He ke algebra of weight k for Γ, i.e. the subring ofEndC(Sk(Γ)) spanned by all He ke operators Tn for n ∈ N. If Γ = Γ0(N) we simply writeTk(N). We use similar notation in other ontexts, too. It is an important theorem that T isfree of nite rank as a Z-module, hen e has Krull dimension one as a ring, and that the map(2.3) HomZ(T, C) → Sk(Γ), φ 7→

∞∑

n=1

φ(Tn)qnwith q = q(z) = e2πiz denes an isomorphism of C-ve tor spa es, whi h is ompatible withthe natural He ke a tion. For any ring A, dene Sk(Γ;A) := HomZ(T, A) equipped with thenatural He ke a tion (i.e. T-a tion), so that we have Sk(Γ) ∼= Sk(Γ; C). We think of elementsin Sk(Γ;A) in terms of formal q-expansions, i.e. as formal power series in A[[q]], by an analogof Eq. 2.3. Note that normalised He ke eigenforms, i.e. those f =∑∞

n=1 anqn ∈ Sk(Γ;A)that satisfy a1 = 1 and Tnf = anf , pre isely orrespond to ring homomorphisms φ : T → Awith φ(Tn) = an. When A is an integral domain, a normalised eigenfun tion gives rise toa prime ideal p of T, namely the kernel of φ. We may think of T/p as the smallest subringof A generated by the an for n ∈ N: the oe ient ring of f in A. Note that Aut(A) a ts onSk(Γ;A) by omposing φ : T → A with σ ∈ Aut(A). Obviously, this a tion does not hangethe ideal p orresponding to an eigenform. Publi ations mathématiques de Besançon - 2011

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78 Asymptoti behaviour of oe ient elds of modular formsWe x a prime number p. We put T := Tk(Γ) := Tk(Γ)⊗Z Zp, Tη := Tk(Γ)⊗Z Qp = T⊗Zp Qpand T := Tk(Γ) := Tk(Γ)⊗ZFp = T⊗ZpFp. Note the isomorphisms Sk(Γ; Zp) ∼= HomZp(T, Zp),Sk(Γ; Qp) ∼= HomZp(T, Qp) and Sk(Γ; Fp) ∼= HomZp(T, Fp) ∼= HomFp(T, Fp).The Gal(Qp/Qp)- onjuga y lasses of normalised eigenforms in Sk(Γ; Qp) (by whi h we meanthe lasses for the Gal(Qp/Qp)-a tion des ribed above) are in bije tion with the prime (andautomati ally maximal) ideals of Tη and also in bije tion with the minimal prime ideals of T,whose set is denoted by MinSpec(T). The se ond bije tion is expli itly given by taking preim-ages for the inje tion T → Tη. Note that T has Krull dimension one, meaning that any primeideal is either minimal (i.e. not ontaining any smaller prime ideal) or maximal. Moreover,the Gal(Fp/Fp)- onjuga y lasses of normalised eigenforms in Sk(Γ; Fp) are in bije tion withSpec(T) = MaxSpec(T). Furthermore, Spec(T) is in natural bije tion with MaxSpec(T) un-der taking preimages for the natural proje tion T ։ T. By a result in ommutative algebra,we have dire t produ t de ompositions

T =∏

m∈MaxSpec(bT)

Tm, T =∏

m∈MaxSpec(bT)

Tm and Tη =∏

p∈Spec(bTη)

Tη,p,where the fa tors are the lo alisations at the prime ideals indi ated by the subs ripts.Denition 2.1. We say that two p1, p2 ∈ MinSpec(T) are ongruent if they lie in thesame maximal ideal m ∈ MaxSpec(T). For p ∈ MinSpec(T), we all T/p the lo al oe ientring and Lp := Frac(T/p) the lo al oe ient eld. We say that p ∈ MinSpec(T) is ramiedif Lp is a ramied extension of Qp. We denote by ip the index of T/p in the ring of integersof Lp. The residue eld T/m ∼= T/m will be denoted by Fm and will be alled the residual oe ient eld.We now establish the onne tion with the usual understanding of the terms in the denition.Let Z ⊂ Q ⊂ C be the algebrai integers and the algebrai numbers, respe tively. As T is ofnite Z-rank, the set of normalised eigenforms in Sk(Γ) is the same as the set of normalisedeigenforms in Sk(Γ; Z). Fix homorphismsZ

ι//

π

==

==

==

==

=Zp

π

Fp

giving rise to Sk(Γ; Z) ι

//

π%%%%

KKKKK

KKKKKK

Sk(Γ; Zp)

π

Sk(Γ; Fp)

.

From this perspe tive, a holomorphi normalised He ke eigenform f =∑∞

n=1 anqn ∈ Sk(Γ)gives rise to an eigenform in Sk(Γ; Fp), alled the redu tion of f modulo p, whose formal q-expansion is π(f) :=∑∞

n=1 π(an)qn ∈ Fp[[q]]. The redu tion orresponds to m ∈ MaxSpec(T)and to m ∈ Spec(T) (we use the same symbol due to the natural bije tion between thetwo sets). The form f also gives rise to an eigenform in Sk(Γ; Qp), whi h orresponds topf ∈ MinSpec(T) and to pf ∈ Spec(Tη) (the same symbol is used again due to the naturalbije tion explained above).Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 79Let g =∑∞

n=1 bnqn be another holomorphi normalised He ke eigenform. If π(f) = π(g), then learly pf ⊂ m and pg ⊂ m, i.e. pf and pg are ongruent. Conversely, let pf , pg ∈ MinSpec(T)su h that pf ⊂ m and pg ⊂ m for some m ∈ MaxSpec(T), so that pf and pg are ongruent.The ideals pf and pg orrespond to Gal(Qp/Qp)- onjuga y lasses in Sk(Γ; Qp) and we an hoose f, g ∈ Sk(Γ; Zp) orresponding to pf and pg with π(f) = π(g). We illustrate thesituation by the diagramT/pf

((((

RRRRRRRRRRRRRRRRRR _

T

8888qqqqqqqqqqqqqqq

&&&&

MMMMMMMMMMMMMMM

f,,

g

22 Zpπ

// // Fp T/m = Fm.? _oo

T/pg

6666llllllllllllllllll

?

OO

Note that f, g an already be found in Sk(Γ; Z) ⊂ Sk(Γ). This justies our usage of the term ongruen e.Moreover, the lo al oe ient ring T/p an be identied with Zp,f := Zp[ι(an)|n ∈ N] andits fra tion eld Lp with Qp,f := Qp(ι(an)|n ∈ N), when e ip is the index of Zp,f in itsnormalisation. Furthermore, the residual oe ient eld, i.e. Fm = T/m, an be interpretedas Fp[π(an)|n ∈ N]. The relation to the arithmeti of the oe ient eld Qf := Q(an|n ∈ N)and the oe ient ring Zf := Z[an|n ∈ N] is apparent.In order to on lude this ba kground se tion, we point out that in the ase k = 2, the oe ient ring Zf is the endomorphism ring of the abelian variety Af atta hed to f . Fromthat point of view, the following analysis an also be interpreted as a study of the arithmeti of the endomorphism algebras of GL2-abelian varieties.3. Semisimpli ity of He ke algebrasWe re all that a nite dimensional ommutative K-algebra, where K is a eld, is semisimpleif and only if it is isomorphi to a dire t produ t of elds (whi h are ne essarily nite eldextensions of K).In this se tion we rst study the semisimpli ity of the He ke algebra Tη. In the ase whenit is semisimple, we relate the non-semisimpli ity of the mod p He ke algebra T to threephonomena: ongruen es between Gal(Qp/Qp)- onjuga y lasses of newforms, rami ationat p of the oe ient elds of newforms and the p-index of the lo al oe ient ring in thering of integers of the lo al oe ient eld.Let f =∑∞

n=1 an(f)qn ∈ Sk(Γ1(M))new be a normalised He ke eigenform and let m be anypositive integer. We dene the C-ve tor spa e Vf (m) to be the span of f(qd) | d | m, wherePubli ations mathématiques de Besançon - 2011

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80 Asymptoti behaviour of oe ient elds of modular formsd runs through all positive divisors of m (in luding 1 and m). Newform theory states that

Sk(Γ1(N)) ∼=⊕

m|N

f∈Sk(Γ1(N/m))new

Vf (m).This is an isomorphism of He ke modules. The He ke operators Tn for (n,m) = 1 restri tedto Vf (m) are s alar matri es with an(f) as diagonal entries. We now des ribe the He keoperator Tℓ on Vf (m) for a prime ℓ. Suppose that ℓr || m. Let ǫ be the Diri hlet hara terof f . Consider the (r + 1) × (r + 1)-matrixA := Af (m, ℓ) :=

aℓ(f) 1 0 0 . . . 0

−δǫ(ℓ)ℓk−1 0 1 0 . . . 0

0 0 0 1 . . . 0... ...0 . . . 0 0 0 1

0 . . . 0 0 0 0

,where δ = 0 if ℓ | (N/m) and δ = 1 otherwise. The He ke operator Tℓ is then given on Vf (m)(for a ertain natural basis) by a diagonal blo k matrix having only blo ks equal to A on thediagonal, where ea h blo k on the diagonal orresponds to a divisor of m/ℓr. Let T be theHe ke algebra of Sk(Γ1(N)) (as in Se tion 2). The algebra TQ := T⊗Z Q is semisimple if andonly if TC := T⊗Z C is semisimple (if and only if Tη is semisimple). By the above dis ussion,TC is semisimple if and only if all the matri es Af (m, ℓ) that appear are diagonalisable.Proposition 3.1. Assume the notation above, M = N/m and k ≥ 2. Moreover, if k ≥ 3assume Tate's onje ture (see [CE, Se tion 1).(a) Assume ℓ ∤ M . Then Af (m, ℓ) is diagonalisable if and only if r ≤ 2.(b) Assume that ℓ | M and that either ℓ || M or that ǫ annot be dened mod M/ℓ. Then

Af (m, ℓ) is diagonalisable if and only if r ≤ 1.( ) Assume that ℓ2 | M and that ǫ an be dened modulo M/ℓ. Then Af (m, ℓ) is diagonalisableif and only if r = 0.Proof. (a) Assume r ≥ 1 (otherwise the result is trivial) and all B the top left 2×2-blo kof A = Af (m, ℓ). The hara teristi polynomial of B is g(X) = X2 − aℓ(f)X + ǫ(ℓ)ℓk−1.We have g(0) 6= 0 and g(X) has dis riminant aℓ(f)2 − 4ǫ(ℓ)ℓk−1, whi h is non-zero, sin e|aℓ(f)| = 2|ℓ|(k−1)/2 would ontradi t [CE, Theorem 4.1. Consequently, A is diagonalisableif and only if apart from B there is at most one more row and olumn.In ases (b) and ( ), note that A is in Jordan form. The result is now immediate, sin e aℓ(f)is non-zero for (b) and zero for ( ) (see [DS, 1.8).We have the immediate orollary (whi h is Theorem 4.2 in [CE).Corollary 3.2. Let N be ubefree and k ≥ 2. If k ≥ 3 assume Tate's onje ture (see [CE,Se tion 1). Then the He ke algebras Tk(Γ1(N)) ⊗ Q and Tk(Γ1(N))η as well as Tk(N) ⊗ Qand Tk(N)η are semisimple.Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 81The prin ipal result of this se tion is the following proposition on the stru ture of the residualHe ke algebra. We assume the notation laid out in Se tion 2, in parti ular, we work with ageneral ongruen e subgroup Γ.Proposition 3.3. Assume that Tη is semisimple (see, e.g., Corollary 3.2), i.e.Tη

∼=∏

p∈MinSpec(bT)

Lp.Then the residual He ke algebra T is semisimple if and only if all of the following three on-ditions are satised:(i) No two p1, p2 ∈ MinSpec(T) are ongruent.(ii) None of the p ∈ MinSpec(T) is ramied.(iii) For all p ∈ MinSpec(T), the index ip = 1.Proof. We rst prove that (i), (ii) and (iii) imply the semisimpli ity of T.The fa t that there is no ongruen e means that in every m ∈ MaxSpec(T) there is a uniquep ∈ MinSpec(Tm). As Tm⊗Zp Qp

∼= Lp, it follows that Tm is a subring of Lp. Due to (iii), Tmis the ring of integers of Lp. Sin e by (ii) Lp is unramied, we get that Tm is the residue eldof the integers of Lp. This shows that T is a produ t of nite elds, i.e. semisimple.Now we prove the onverse dire tion and assume that T is semisimple. Let m ∈ MaxSpec(T).Let p1, p2, . . . , pm ∈ MinSpec(T) be the distin t minimal primes ontained in m. Then Tm⊗Zp

Qp∼= Lp1 × · · · ×Lpm . Due to the non-degeneration Tm

∼= Tm⊗Zp Fp∼= Fpn for some n. Sin e

dimQpTm ⊗Zp Qp = n, we have [Lpi

: Qp] ≤ n for i = 1, . . . ,m.Let Oi be the ring of integers of Lpifor i = 1, . . . ,m. It ontains Tm/pi with index ipi

.Tensoring the exa t sequen e of Zp-modules0 → Tm/pi → Oi → Oi/(Tm/pi) → 0with Fp over Zp we obtain the exa t sequen e of Fp-ve tor spa es:

Fpn → Oi ⊗Zp Fp → (Oi/(Tm/pi)) ⊗Zp Fp → 0.Sin e the map on the left is a ring homomorphism, it is inje tive. Now dimFp Oi ⊗Zp Fp ≤ nimplies that Oi is unramied and that ipi= 1. Thus [Lpi

: Qp] = n for i = 1, . . . ,m and,hen e, m = 1, on luding the proof.4. Observations and QuestionsIn this se tion, we explain and expose our omputer experiments and we ask some ques-tions suggested by our studies. All omputer al ulations were performed using Magma (see[BCP). Publi ations mathématiques de Besançon - 2011

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82 Asymptoti behaviour of oe ient elds of modular forms4.1. Semisimpli ity of the residual He ke algebra. A lo al nite-dimensional om-mutative Fp-algebra A is semisimple if and only if it is simple, whi h is equivalent to A beingeld. We take the dimension of the maximal ideal m of A as a measure for the deviation of Afrom being semisimple. In parti ular, A is a eld if and only if m has dimension 0.For given prime p, level N and weight k we study the sum of the residue degrees of all primeideals:a

(p)N,k =

m∈Spec(Tk(N))

[Fm : Fp].Clearly, a(p)N,k is less than or equal to the Fp-dimension of Sk(N ; Fp). Hen e, Tk(N) is semisim-ple if and only if a

(p)N,k is equal to this dimension.We intend to study the asymptoti behaviour of the fun tion a

(p)N,k for a xed prime p and xedweight k as a fun tion of the level N . For simpli ity, we let N run through the prime numbersonly in order to avoid ontributions from lower levels via the degenera y maps. We shouldpoint out that there an be ontributions from lower weights: an eigenform in Sk(N ; Fp) alsolives in Sk+n(p−1)(N ; Fp) for all n ≥ 0 by multipli ation by the Hasse invariant. Note that for

p > 2 and k = 2, as well as for p > 3 and k = 4 there is no su h ontribution.Our omputational ndings are best illustrated by plotting graphs. In ea h of the follow-ing plots, the prime p and the weight k are xed and on the x-axis we plot d(N) :=

dimFp

Sk(N ; Fp) and on the y-axis the fun tion a(p)N,k as a fun tion of N , i.e. ea h N givesrise to a dot at the appropriate pla e. The straight line in the graphs was determined as thelinear fun tion α · d(N) whi h best ts the data (a ording to gnuplot and the least squaresmethod).In the weights that we onsidered we observed a behaviour for p = 2 whi h seems to be ompletely dierent from the behaviour at all other primes. We made plots for all odd primesless than 100 and weight 2 and present a sele tion here. The graphs that we leave out lookvery similar.Figure 1

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Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 83Figure 3 0

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Publi ations mathématiques de Besançon - 2011

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84 Asymptoti behaviour of oe ient elds of modular formsFigure 9 0

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In Figures 110 the levels range over all primes up to a ertain bound (whi h is not the samefor all p). We observe that non-semisimpli ity seems to be a rather rare phenomenon whi hbe omes rarer for growing p, as one might have guessed. In the next gures, we analyse the ases p = 3, 5 still for weight 2 more losely by letting the levels range through all primesbetween 3000 and 10009 subdivided into four onse utive intervals.Figure 11 240

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Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 85Figure 15 240

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One an observe that the slope of the best tting line through the origin seems to be in reasingwith growing dimension. Although we only omputed relatively litte data, we in lude twoexamples for weight 4. They do not suggest any signi ant dieren e to the weight 2 ase.Figure 19 0

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We are led to ask the following question. Publi ations mathématiques de Besançon - 2011

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86 Asymptoti behaviour of oe ient elds of modular formsQuestion 4.1. Fix an odd prime p and an even k ≥ 2. Let a(N) := a(p)k,N and d(N) :=

dimFp

Sk(N ; Fp). Does the following statement hold?For all ǫ > 0 there is Cǫ > 0 su h that for all primes N the inequalitya(N) > (1 − ǫ)d(N) − Cǫholds.We ontrast the situation, whi h seems very similar for every odd prime, with the one for

p = 2 and k = 2. We do not onsider any higher weights due to the ontributions fromweight 2, whi h would `disturb' the situation. The following plots take prime numbers N intoa ount that lie in six dierent intervals up to 12000.Figure 21 0

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Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 87Figure 25 320

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In spite of the very irregular behaviour, it is remarkable that the slope of the best tting linethrough the origin is always just a little bigger than 12 .At the moment we annot fully explain this behaviour. Contributions from weight one playsome role. However, probably more important are ongruen es of forms having Atkin-Lehnereigenvalue +1 with forms having eigenvalue −1. As Johan Bosman observes in a remarkin [B, it follows from the onne tedness of the spe trum of the He ke algebra ([M, 10.6) for

k = 2 and prime levels that there is always at least one su h ongruen e for p = 2, wheneverthe +1- and the −1-eigenspa e are nonempty.We are led to ask the following question.Question 4.2. Fix an even weight k ≥ 2. Let a(N) := a(2)k,N and d(N) := dim

F2Sk(N ; F2).Are there 1 > α ≥ β > 0 and onstants C,D > 0 su h that the inequality

α · d(N) + C > a(N) > β · d(N) − Dholds?4.2. Average Residue Degree. We now study the average residue degree, whi h wedene for given level N , weight k and prime p asb(p)N,k =

1

# Spec(Tk(N))

m∈Spec(Tk(N))

[Fm : Fp] =a

(p)N,k

# Spec(Tk(N)).We made omputations for weight 2 and all primes p less than 100, where N runs throughthe same ranges as previously. We again plot the dimension d(N) on the x-axis and thefun tion b

(p)N,k on the y-axis and the straight line is again the best tting fun tion α · d(N),although we believe that this is not the right fun tion to take (see below). We again presenta sele tion of prime numbers as before, however, in luding p = 2 from the beginning. Thegraphs that we leave out have very similar shapes.

Publi ations mathématiques de Besançon - 2011

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88 Asymptoti behaviour of oe ient elds of modular formsFigure 27 0

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Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 89Figure 33 0

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Here are again two examples for weight 4.Figure 37 0

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Very roughly speaking the data suggest that the average residue degree grows with the di-mension, as is ertainly to be expe ted. We also ondu ted a loser analysis for the primes 2,3 and 5. For p = 2 we used all primes in dierent intervals up to 12000 and obtained theseplots: Publi ations mathématiques de Besançon - 2011

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90 Asymptoti behaviour of oe ient elds of modular formsFigure 39 15

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x * 0.055893

Here are the plots for p = 3, 5 and the primes between 3000 and 10009 subdivided into fourintervals.Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 91Figure 44 20

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92 Asymptoti behaviour of oe ient elds of modular formsFigure 50 30

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idue

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ree

Dimension k = 2, p = 5

x * 0.080942 Figure 51 40

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rage

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idue

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ree

Dimension k = 2, p = 5

x * 0.077091

We observe that the slope of the best tting line goes slightly down with the dimension.This strongly suggests that taking a straight line does not seem to be quite orre t. We alsomade logarithmi plots, whi h we do not reprodu e here; they seemed to suggest to us thata behaviour of the form b(p)N,k ∼ const · d(N)α is not quite orre t either (the best hoi e of αseems to be lose to 1 in a ordan e with the previous dis ussion).These omputations suggest the following question.Question 4.3. Fix a prime p and an even weight k ≥ 2. Let b(N) := b

(p)k,N and d(N) :=

dimFp

Sk(N ; Fp). Do there exist onstants C1, C2 and 0 < α ≤ β < 1 su h that the inequalityC1 + α

d(N)

log(d(N))≤ b(N) ≤ C2 + β · d(N)holds?We remark that if a

(p)N,k behaves like d(N), as suggested by Question 4.1, then Question 4.3 isequivalent to asking that # Spec(Tk(N)) does not grow faster than a onstant times log(d(N)).The phenomenon that for odd primes p most of the dots in the diagrams seem to lie on orvery lose to ertain distinguished lines through the origin is natural in view of Question 4.1:the slope of the line on or lose to whi h a dot lies is just 1

#Spec(Tk(N)).4.3. Maximum Residue Degree. Now we turn our attention to the maximum residuedegree, whi h we dene for given level N , weight k and prime p as

c(p)N,k = max[Fm : Fp] | m ∈ Spec(Tk(N)).We made omputations for weight 2 and all primes p less than 100, where N runs throughthe same ranges as previously. We again plot the dimension d(N) on the x-axis and thefun tion c

(p)N,k on the y-axis and the straight line is again the best tting fun tion α · d(N).This time we believe that this might be the right fun tion to take. Here is again a sele tionof the plots that we obtained.

Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 93Figure 52 0

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Figure 54 0

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x * 0.387289

Figure 56 0

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x * 0.391898 Figure 57 0

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idue

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ree

Dimension k = 2, p = 31

x * 0.383301

Publi ations mathématiques de Besançon - 2011

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94 Asymptoti behaviour of oe ient elds of modular formsFigure 58 0

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x * 0.384707 Figure 59 0

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x * 0.416753

Figure 60 0

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x * 0.405110 Figure 61 0

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x * 0.393488

Here are again two examples for weight 4.Figure 62 0

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x * 0.362600 Figure 63 10

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idue

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ree

Dimension k = 4, p = 11

x * 0.377393

The data ertainly suggest that the maximum residue degree grows with the dimension. It isremarkable to see that the slopes of the best tting lines all seem to be very lose to ea h other with the single ex eption of the ase p = 2, whi h might be aused by the same phenomenonas earlier. Also in this ase we ondu ted a loser analysis for the primes 2, 3 and 5. Forp = 2 we used all primes in dierent intervals up to 12000 and obtained these plots:Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 95Figure 64 60

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x * 0.289400 Figure 67 100

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Figure 68 150

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x * 0.276529

Here are the plots for p = 3, 5 and the primes between 3000 and 10009 subdivided into fourintervals.Publi ations mathématiques de Besançon - 2011

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96 Asymptoti behaviour of oe ient elds of modular formsFigure 69 40

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x * 0.374745

Figure 71 100

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Figure 73 60

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Publi ations mathématiques de Besançon - 2011

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Mar el Mohyla and Gabor Wiese 97Figure 75 100

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x * 0.366508

We observe that, although the best tting lines were omputed using dierent intervals, theirslopes are very lose to ea h other. These omputations suggest the following question.Question 4.4. Fix a prime p and an even weight k ≥ 2. Let c(N) := c(p)k,N and d(N) :=

dimFp

Sk(N ; Fp). Do there exist onstants C1, C2 and 0 < α ≤ β < 1 su h that the inequalityC1 + α · d(N) ≤ c(N) ≤ C2 + β · d(N)holds? Referen es[BLGHT T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor. A family of Calabi-Yau varietiesand potential automorphy II. To appear in P.R.I.M.S.[BCP W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The user language. J.Symb. Comp. 24(34) (1997), 235265.[B J. Bosman. Modular forms applied to the omputational inverse Galois problem. Preprint, 2010.[CHT L. Clozel, M. Harris and R. Taylor. Automorphy for some l-adi lifts of automorphi mod lrepresentations. Pub. Math. IHES 108 (2008), 1181.[CE Coleman, Robert F.; Edixhoven, Bas. On the semi-simpli ity of the Up-operator on modularforms. Math. Ann. 310 (1998), no. 1, 119127.[DiWi L. Dieulefait, G. Wiese. On Modular Forms and the Inverse Galois Problem. A epted forpubli ation in the Transa tions of the AMS.[DS Deligne, Pierre; Serre, Jean-Pierre. Formes modulaires de poids 1. Ann. S i. É ole Norm. Sup.(4) 7 (1974), 507530.[DI F. Diamond and J. Im.Modular forms and modular urves, in Seminar on Fermat's Last Theorem(Toronto, ON, 19931994), 39133, Amer. Math. So ., Providen e, RI, 1995.[HSHT M. Harris, N. Shepherd-Barron and R. Taylor. A family of Calabi-Yau varieties and potentialautomorphy. Annals of Math. 171 (2010), 779813.[M B. Mazur. Modular urves and the Eisenstein ideal. Inst. Hautes Études S i. Publ. Math. No. 47(1977), 33186 (1978).[S J.-P. Serre. Répartition asymptotique des valeurs propres de l'opérateur de He ke Tp. J. Ameri anMathemati al So iety 10(1), 1997, 75102. Publi ations mathématiques de Besançon - 2011

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98 Asymptoti behaviour of oe ient elds of modular forms[T R. Taylor. Automorphy for some l-adi lifts of automorphi mod l representations. II Pub. Math.IHES 108 (2008), 183239.[W G. Wiese. On proje tive linear groups over nite elds as Galois groups over the rational numbers.In: 'Modular Forms on S hiermonnikoog' edited by Bas Edixhoven, Gerard van der Geer and BenMoonen. Cambridge University Press, 2008, 343350.23 o tobre 2010Mar el Mohyla, Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstraÿe 29,45326 Essen, GermanyGabor Wiese, Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstraÿe 29, 45326Essen, Germany • E-mail : gabor.wieseuni-due.de

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OPTIMAL CURVES OF GENUS 1, 2 AND 3byChristophe RitzenthalerAbstra t. In this survey, we dis uss the problem of the maximum number of points of urves of genus 1, 2 and 3 over nite elds.Résumé (Courbes optimales de genre 1, 2 et 3). Nous examinons la question dunombre maximum de points pour les ourbes de genre 1, 2 et 3 sur les orps nis.1. Introdu tionThe foundations of the theory of equations over nite elds were laid, among others, bymathemati ians like Fermat, Euler, Gauss and Ja obi (see [Di 66). Subsequently, there waslittle a tivity in the eld at least until the end of the 19th entury and the study of the zetafun tion of a urve. Initiated by Dedekind, Weber, Artin and S hmidt, this work led to ananalogue of the Riemann hypothesis whi h was proved by Hasse in the ase of ellipti urvesand then by Weil in general in 1948 (see [Wei48). The third, modern, period starts in 1980with the work of Goppa [Gop77, Gop88. His onstru tion of error- orre ting odes withgood parameters from urves over nite elds renewed the interest in this theory.With this appli ation in mind, the theory has fo used on the maximum number of points of a(proje tive, geometri ally irredu ible, non singular) urve of genus g over a nite eld k = Fq,denoted Nq(g). Asymptoti results, i.e. values of Nq(g)/g when g goes to innity and q isxed, drew attention rst, but Serre, in his le tures at Harvard [Ser85, gave equal treatmentto the `dual' ase, i.e. values of Nq(g) when g is xed and q varies. It qui kly appeared thatdetermining Nq(g) was a hard problem and as soon as g ≥ 3, only sparse values are known(see for instan e the web page www.manypoints.org for the best estimates when q is small).In this survey, we are going to des ribe the main ideas that have been developed to deal with2000 Mathemati s Subje t Classi ation. Primary 11G20, 11G10 Se ondary 14K25, 14H45.Key words and phrases. Optimal urve, isogeny lass, inde omposable polarization, hermitian module,Serre's obstru tion, plane quarti , Siegel modular form, Hasse-Weil-Serre bound.The author a knowledges partially supported by grant MTM2006-11391 from the Spanish MEC and by grantANR-09-BLAN-0020-01 from the Fren h ANR..

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100 Optimal urves of genus 1, 2 and 3the ases 1 ≤ g ≤ 3. It is interesting to note that for ea h value of g, we will be onfronted notonly to harder omputations but also to a ompletely new kind of issue. In order to emphasizethis progression, we will not onsider the a tual value of Nq(g) but the following sub-problem.As we shall re all in Se tion 2.1, Nq(g) ≤ 1 + q + g⌊2√q⌋ and we an wonder when Nq(g)rea hes this bound. If it does, a urve with this number of points is alled optimal and weare going to ask for whi h values of q su h urves exist.When g ≤ 3, the lassi al game to prove or disprove the existen e of optimal urves is1. to prove the existen e (or not) of an abelian variety A/k with a `good' Weil polynomial(Se tion 2). This is going to ontrol the number of points on a possible urve C/k su hthat Jac C ≃ A.2. to put a good polarization a on A su h that (A, a)/k is geometri ally (i.e. over k) theJa obian of a urve C with its anoni al polarization (Se tion 3).3. to see if C admits a model C/k su h that (Jac C, j) ≃ (A, a) (where j is the anoni alpolarization of C). We will see that if C is non hyperellipti , there an be an obstru tionto this des ent (see Se tion 4) and for g = 3, we will propose solutions to address the omputation of the obstru tion (see Se tion 5).Most ideas we are going to present here are already ontained in [Ser85 but our proofs forg = 1 and 2 are sometimes dierent from the original ones and take advantage of subsequentsimpli ations of the theory.Conventions and notation. In the following g ≥ 1 is an integer and q = pn with p aprime and n > 0 an integer. The letter k denotes the nite eld Fq and K any perfe t eld.When we speak about a genus g urve we mean that the urve is proje tive, geometri allyirredu ible and non-singular. If A and B are varieties over a eld K, when we speak of amorphism from A to B we always mean a morphism dened over K. So, for instan e End(A)is the ring of endomorphisms dened over K, A ∼ B means A isogenous to B over K, et . If(A, a) and (B, b) are polarized abelian varieties, by an isomorphism between them, we alwaysmean `as polarized abelian varieties'.A knowledgements. I would like to thank Christian Maire for suggesting me to writethis survey. This is part of my `habilitation' thesis [Rit09 whi h was defended during theworkshop Theory of Numbers and Appli ations whi h was organized by Karim Belabas andChristian Maire in Luminy in De ember 2009. I am really grateful to Detlev Homann forthe referen es of Remark 3.5 and to the Number Theory List ommunity and parti ularly toSamir Siksek for helping me with Remark 3.10.2. Control of the isogeny lass2.1. Bounds. Let C/k be a genus g urve. We re all that its Weil polynomial χC is theWeil polynomial of JacC/k, i.e. the hara teristi polynomial of the a tion of the k-FrobeniusPubli ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 101endomorphism on an ℓ-adi Tate module for any prime ℓ 6= p. It is well known that it an bewrittenχC =

g∏

i=1

(X2 + xiX + q) ∈ Z[X]with xi ∈ R and |xi| ≤ 2√

q. Sin e#C(k) = q + 1 +

g∑

i=1

xi,it is lear that #C(k) ≤ 1 + q + ⌊2g√q⌋, whi h is known as Hasse-Weil bound [Wei48 andso Nq(g) is less than this bound too. It is possible to improve this bound as the followinglemma shows.Lemma 2.1 (Hasse-Weil-Serre bound [Ser83b). Let m = ⌊2√q⌋. ThenNq(g) ≤ 1 + q + gm.Proof. It is enough to use the arithmeti -geometri mean inequality:

1

g

g∑

i=1

(m + 1 − xi) ≥(

g∏

i=1

(m + 1 − xi)

)1/g

≥ 1,the last inequality oming from the fa t that the produ t is a non-zero integer.This motivates us to give the following denition.Denition 2.2. We say that a genus g urve C/Fq is optimal if#C(Fq) = q + 1 + gm.In that ase Nq(g) = q + 1 + gm.Note that the previous denition is not universally a epted. Some authors all maximal(or Fq-maximal) what we all optimal by referen e to the histori al ases with n even and

Nq(g) = q + 1+ gm. We prefer to keep the word maximal for urves whi h numbers of pointsis equal to Nq(g) and our terminology is oherent with the histori al one as well.Remark 2.3. If g ≥ (q − √q)/2, the bound an be improved, thanks to the expli itmethods of Oesterlé (and is known as Oesterlé bound [Ser83b). It uses the fa t that thenumber of pla es of ea h degree on the urve is non negative. As we will mainly deal withsmall values of g ompared to q, the Hasse-Weil-Serre bound will be our referen e.Publi ations mathématiques de Besançon - 2011

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102 Optimal urves of genus 1, 2 and 32.2. Existen e of the isogeny lass. Equality in the arithmeti -geometri mean in-equality is equivalent to the fa t that all terms in the sum are equal and so xi = m for all1 ≤ i ≤ g. Hen e, if an optimal urve C exists, its Weil polynomial has the parti ular simpleexpression

χC = (X2 + mX + q)g.Honda-Tate theory as explained in [Tat66, [Hon68, [Wat69, [MW71 or [Tat71 showsthat if p ∤ m (resp. n is even) then Jac C is isogenous to Eg where E is an ordinary (resp.supersingular) ellipti urve with tra e −m. However, if p|m and n is odd, this might not betrue (see the proof of Proposition 2.5 below) and there is for instan e a simple abelian varietyof dimension 9 over F59 with su h Weil polynomial. If we restri t to g ≤ 3, it an be provedthat this never happens (see for instan e the proof of Corollary 4.2 of [NR10).Lemma 2.4. If C/Fq is an optimal urve of genus g ≤ 3 then Jac C is isogenous to Egwhere E is an ellipti urve of tra e −m.The rst ne essary ondition is then to see whether su h an ellipti urve exists or not.Proposition 2.5 (Deuring [Deu41). There does not exist an ellipti urve with tra e−m if and only if n ≥ 3, n is odd and p|m.Proof. Let F = X2 + mX + q. Sin e m < 2

√q if and only if q is not a square, F isirredu ible over Q when n is odd and F = (X +

√q)2 when n is even.If n is odd, by [Wat69, p.527, the minimal e for whi h χ = F e is the Weil polynomial ofan abelian variety of dimension e over k is the least ommon denominator of vp(Fν(0))/nwhere Fν denotes the fa tors of F in Qp[t] and vp the p-adi valuation of Qp. Hen e F is theWeil polynomial of an ellipti urve if and only if n|vp(Fν(0)) for all fa tors. This is of oursesatised if n = 1. Looking at the Newton polygon of F , we see that if p ∤ m then vp(Fν(0)) = nor 0, so e = 1. With the same te hnique, if n > 1 odd and p|m, then vp(Fν(0)) < n and so

e > 1.If n is even, we apply the previous arguments to F = X +√

q. Sin e vp(√

q)/n = 1/2, e = 2so F 2 = X2 + mX + q is the Weil polynomial of an ellipti urve.A tually, the only values of q = p for whi h p|m are q = 2 or q = 3.Remark 2.6. For any value of −m ≤ t ≤ m, one knows if an ellipti urve with tra et exists (see [Wat69, Th.4.1). Also, the possible Weil polynomials of the isogeny lasses ofabelian surfa es (resp. threefolds) an be found in [MN02, Lem.2.1,Th.2.9 (resp. [Xin96,Hal10). 3. Existen e of an inde omposable prin ipal polarizationThe Ja obian of a genus g urve C/K is naturally equipped with a prin ipal polarization jindu ed by the interse tion pairing on the urve C. Sin e the theta divisor Symg−1 C → Jac Casso iated to j is geometri ally irredu ible, (Jac C, j) is geometri ally inde omposable, i.e.Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 103there does not exists an abelian subvariety B ⊂ Jac C dened over K su h that j indu es onB a prin ipal polarization. Conversely, starting with A = Eg where E is an ellipti urve, itis lear that A always admits a prin ipal polarization a0 given by the produ t of the prin ipalpolarizations on ea h fa tor. As a0 is de omposable, (A, a0) is not (even geometri ally) aJa obian. Hen e `good' prin ipal polarizations on A (or on abelian varieties in the isogeny lass of A) have to be more subtle. Lu kily, equivalen es of ategory have been developed totranslate the existen e of an inde omposable polarization into the existen e of purely algebrai obje ts. As far as I know several points of view o-exist and it is not lear to see how to gofrom one to the other. I shall use Serre's one and mention others in remark.Remark 3.1. Howe [How95,[How96 has developed a powerful ma hinery to prove theexisten e of a prin ipally polarized abelian variety in the isogeny lass of an abelian varietyA/k. But only when A is simple, it is easy to see that the polarization is inde omposable (see[Ryb08 for the ase E×B where E is an ellipti urve and B a geometri ally simple abeliansurfa e).3.1. The equivalen es. Let us start with E ordinary. Let E/k be an ordinary ellipti urve with tra e t. If π denotes the Fq-Frobenius endomorphism of the urve E, then thering R := Z[X]/(X2 − tX + q) is isomorphi to Z[π] ⊂ End(E). Serre [Ser85, Se.50-53,[Lau02, Appendix denes an equivalen e of ategory T between the ategory of abelianvarieties whi h are isogenous to a power of E and R-modules of nite type without torsion.The fun tor T maps an obje t A to the R-module L = Hom(E,A). Obviously, the rank of Lis equal to the dimension of A. This fun tor also behaves ni ely with respe t to duality: if wedenote L the ring of anti-linear homomorphism f : L → R (i.e. f(rx) = rf(x) for all r ∈ Rand x ∈ L) then T (A) = L. Thus a morphism a : A → A denes a morphism h : L → L andhen e an hermitian form H : L×L → R. Serre proves that a is a polarization if and only if His positive denite, that a is prin ipal if (L,H) is unimodular (i.e. h(L) = L) and moreover(geometri ally) inde omposable if and only if (L,H) is inde omposable, i.e. annot be writtenas a sum of orthogonal sub-modules. The ouple (L,H) is alled a hermitian module.Remark 3.2. This equivalen e is inspired by the lassi al theory over C, whi h is notsurprising sin e ordinary abelian varieties an be lifted anoni ally and this is used in [Del69.When A = Eg and End(E) ≃ R, a more expli it point of view an be onsidered looking atthe hermitian matrix M := a−1

0 a ∈ End(A) = Mg(End(E)) ≃ Mg(R) (see [Rit10, [Lan06).For g = 2, Kani's onstru tion [Kan97 also gives ne essary and su ient onditions for`gluing' two ellipti urves along their n-torsion for n > 1. Both points of view are related bythe Cholewsky de omposition of M .A lassi ation of rank 2 and 3 hermitian modules was a hieved in [Hof91, Th.8.1,8.2 (seealso [S h98 for further omputations) and translates into the following result.Publi ations mathématiques de Besançon - 2011

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104 Optimal urves of genus 1, 2 and 3Proposition 3.3. Let E be an ordinary ellipti urve with tra e t. There is no abeliansurfa e (resp. threefold) with a geometri ally inde omposable prin ipal polarization in the lassof E2 (resp. E3) if and only if t2 − 4q ∈ −3,−4,−7 (resp. t2 − 4q ∈ −3,−4,−8,−11).Remark 3.4. For g = 2, the result an be tra ed ba k to [HN65, p.14, where the authorsprove the existen e of genus 2 urves whi h Ja obian is isomorphi to E2 by onstru tingfree inde omposable hermitian modules (in [Hay68, the pre ise number of isomorphism lasses of su h urves is omputed). For g = 2 or 3, it ould also be dedu ed from themass formulae (i.e. number of weighted lasses by the order of their automorphism group) of[HK86, HK89 (although, a ording to Homann (lo . it. p.400) there is a minor mistakein these omputations).Remark 3.5. For g > 3, there have been several partial answers on the existen e ofinde omposable unimodular positive denite hermitian modules of rank g over the ring ofintegers of an imaginary quadrati eld Q(√−d). It seems that in [Zhu97 and [WL01 a omplete answer is given: there always exists one, ex ept when d = 1 and g = 5 or d = 3and g = 4, 5, 7. One should be areful sin e, a ording to the Maths inet review of [Zhu97by Homann, the proofs ontain several mistakes. Also, I do not know if the ase of nonmaximal orders has been onsidered.Assume now that E is supersingular. More pre isely, let E/Fp be an ellipti urve with tra e

0, so that E is supersingular, all the geometri automorphisms of E are dened over Fp2 andTr(E/Fp2) = −2p = −m. One says that an abelian variety A (resp. a urve C) is superspe ialif A (resp. Jac C) is geometri ally isomorphi to a produ t of supersingular ellipti urves. Aresult of Deligne (see [Shi79, Th.3.5) shows that when g > 1, a superspe ial abelian varietyof dimension g is geometri ally isomorphi to Eg (whereas for g = 1 there are non-isomorphi supersingular ellipti urves as soon as p > 7). However, the des ription of the isogeny lass ismade more ompli ated than in the ordinary ase by the existen e of ` ontinuous' families ofisogenies. For instan e, already when g = 2 (see [Oor75), a supersingular abelian surfa e iseither geometri ally isomorphi to E2 (and so superspe ial) or of the form E2/αp where αp isthe unique lo al-lo al group s heme over Fp, the inje tion of αp in E2 being parametrized byP1(Fp) \P1(Fp2). In the latter, it an be shown that A is not superspe ial and the des riptionof the polarizations on this obje t is more evolved. For this reason, we will on entrate onlyon existen e results and limit ourselves to the superspe ial ase.Remark 3.6. Note that it is still possible to obtain a omplete des ription for g = 2 inthe non-superspe ial ase like in [IKO86 or [HNR09, Part.2 where the mass formula of[Ibu89 were used.As in Remark 3.2, we des ribe the polarizations on A = Eg by matri es M := a−1

0 a inEnd(A) = Mg(End(E)). Now, End(E) is a quaternion algebra, so we need results on thenumber ng of positive denite quaternion hermitian forms. Then, to obtain the number of(geometri ally) inde omposable polarizations on Eg, the idea is to subtra t to ng the numberPubli ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 105of polarizations oming from ombinations of lower dimensional abelian varieties. In this way,one getsProposition 3.7 ([Eke87, Prop.7.5). There is no geometri ally inde omposable prin i-pal polarization on Eg if and only if g = 2 and p = 2 or 3, or g = 3 and p = 2.Remark 3.8. More pre isely, Ekedahl gives in [Eke87, Prop.7.2 the mass of inde ompos-able prin ipal polarizations on Eg. However, Bro k [Bro93, Th3.10. orre ts a mistake inthe ase g = 3. He also ompletes and re overs several results obtained in [HI83, I, [KO87for g = 2 and in [Has83, [Oor91 for g = 3. For instan e, in [Bro93, Th.3.14,Th.3.15,he gives the number of genus 2 and genus 3 superspe ial urves for ea h possible group ofautomorphisms.3.2. Appli ation. We an now answer the question of the existen e of a good polariza-tion when g ≤ 3.Theorem 3.9. Let E be an ellipti urve with tra e −m. There is no abelian surfa e(resp. threefold) with a geometri ally inde omposable prin ipal polarization in the isogeny lass of E2 (resp. E3) if and only if q = 4 or 9 or m2 − 4q ∈ −3,−4,−7 (resp. q = 4 or16 or m2 − 4q ∈ −3,−4,−8,−11).Proof. When p ∤ m, E is ordinary so we an use Proposition 3.3.When n is odd and p|m, E exists if and only if q = 2 or 3, whi h leaves these two ases to betreated apart (for instan e by extensive omputer resear h of urves using Theorem 4 or byRemark 3.10).When n is even then p|m. We distinguish several ases. When p > 3 and g = 2 (resp. p > 2 and g= 3), Proposition 3.7 shows that there isalways an inde omposable prin ipal polarization on E2 (resp. E3). Note than when 4|n,the present E is the quadrati twist of the ellipti urve in Proposition 3.7. When p = 2 and g = 2, expli it onstru tions as in [MN07 or a `gluing' argument as in[Ser85, Se.32, [Sha01, Prop.30 shows that one an get a urve C/F2n su h that Jac Cis isogenous to E2 as soon as n > 2. When p = 2, g = 3 and n > 4, An expli it non hyperellipti urve C/F2n su h that

Jac C ∼ E3 an be onstru ted. (see [Rit09, Lem.2.3.8). Note that in [NR08, themore general question of the existen e of a Ja obian in the isogeny lass of a supersingularabelian threefold in hara teristi 2 is addressed. Finally when p = 3 and g = 2, one an nd an expli it onstru tion in [Kuh88 (seealso [Sha01, Cor.37 where a mistake is orre ted) as soon as n > 2. This work was alsogeneralized to all supersingular abelian surfa es in hara teristi 3 in [How08.Remark 3.10. The ases m2−4q ∈ −3,−4 an also be ex luded thanks to a proof dueto Beauville [Sha01, Th.16, [Ser85, Se.13 without any hypothesis on the p-rank of E (andPubli ations mathématiques de Besançon - 2011

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106 Optimal urves of genus 1, 2 and 3then q = 2, 3 are overed).As onje turally, there is innitely many p in the forms p = x2 + 1 and p = x2 + x + 1 theequations m2 − 4pn ∈ −3,−4 have innitely many solutions with n = 1. For n > 1 odd,one knows that the set of solutions is nite. For instan e, in [Ser83a, we nd that there isonly one solution to the equation q = x2 + x + 1 namely q = 73 and none to q = x2 + 1.Similarly, the ase of dis riminant −7 orresponds to the equation q = x2 +x+2 with uniquesolutions q ∈ 23, 25, 213 when n > 1 is odd.The ase of dis riminant −8 orresponds to q = x2 + 2, whi h, when n > 1 is odd, has q = 33for unique solution. This is proved using the same arguments as the last ase below.Finally, the ase of dis riminant −11 orresponds to q = x2 + x + 3. When n > 1 is odd,q = 35 is the unique solution thanks to the following argument due to Samir Siksek. Theequation an be rewritten (2x + 1)2 + 11 = 4pn and fa tors in K = Q(

√−11) as

(

2x + 1 +√−11

2

)(

2x + 1 −√−11

2

)

= pn.Sin e n is odd and OK is a prin ipal domain, there exists α = (a + b√−11)/2 ∈ OK su hthat αn = (2x + 1 +

√−11)/2 and αβ = p with β = α. Now, note that αn − βn =

√−11 andsin e (αn − βn)/(α − β) = 1/b ∈ OK , we see that b = ±1. Hen e, if we x n, we an nd thenite set of integer solutions of this polynomial equation in a. However, to solve it for all nwe have to invoke the mu h deeper theorem from [BHV01 whi h tells us that if there is asolution then n < 4, n = 5 or n = 12. Indeed, with the terminology and notation of lo . it.,one sees that un = (αn − βn)/(α − β) = ±1 is a Lu as number without primitive divisor, sothe Lu as pair (α, β) is n-defe tive. 4. Optimal urvesAs we have seen in Se tion 3, the strategy we have applied so far works in any dimension. Ifwe now have to restri t ourselves to the dimensions less than or equal to 3 is be ause, in these ases, the ondition `has an inde omposable prin ipal polarization' is geometri ally su ientto be the Ja obian of a urve. This is not true when the dimension is bigger, as it is provedsimply by noting that the dimension of the moduli spa e of urves of genus g, 3g − 3, is lessthan the dimension of the moduli spa e of prin ipally polarized abelian varieties of dimension

g, g(g + 1)/2. However,Proposition 4.1 ([OU73). For g ≤ 3, any geometri ally inde omposable prin ipally po-larized abelian variety (A, a)/K is the Ja obian of a urve C over K.So, given (A, a)/K as in Proposition 4.1, the question boils down to know whether one andes end the urve C to a urve C over K su h that (Jac C, j) ≃ (A, a). Surprisingly theanswer is `not always'.Theorem 4.2 (Arithmeti Torelli theorem). There is a unique model C/K of C su hthat:Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 1071. If C is hyperellipti , there is an isomorphism(Jac C, j)

∼−−−−→ (A, a).2. If C is not hyperellipti , there is a unique quadrati hara ter ε of Gal(K/K), and anisomorphism(Jac C, j)

∼−−−−→ (A, a)εwhere (A, a)ε is the quadrati twist of A by ε.Remark 4.3. It is tri ky to nd the right origin of the previous result. In [Ser85, Se.69,Gouvéa indi ates `Oort +. . . ' as a referen e. One an indeed nd in [Oor91, Lem.5.7 asimilar result (but this is 1991). Sekigu hi also worked on this question but after two errata,he gives in [Sek86 only the existen e of the model C/K but does not speak about ε. One an also nd this result in [Maz86, p.236.The notation (A, a)ε = (Aε, aε) should be understood as follows. The variety Aǫ is uniquelydened up to isomorphism by the following property: there exists a quadrati extension L/Kand an isomorphism φ : A → Aǫ dened over L su h that for all σ ∈ Gal(K/K) one hasφσ = ε(σ)φ. The polarization aε is the pull-ba k of a by φ−1.This result is a onsequen e of Weil's des ent as explained in [Ser68, 4.20 and of Torellitheorem [Mat58, p.790-792. The s hism whi h appears between the hyperellipti and nonhyperellipti ase is due to the fa t that

Aut(Jac C, j) ≃

Aut(C) if C is hyperellipti ,Aut(C) × ±1 if C is non hyperellipti .Denition 4.4. The hara ter ε (or the dis riminant of the extension L/K) is alledSerre's obstru tion. By extension, in the hyperellipti ase, we say that ε is trivial.Let us emphasize why this obstru tion is an issue in our strategy. So far we have been able toprove in ertain ases the existen e of a geometri ally inde omposable prin ipally polarizedabelian variety (A, a)/k with Weil polynomial (X2 + mX + q)g. Thanks to Proposition 4.1,we know that it is geometri ally the Ja obian of a urve C. If the obstru tion is trivial, then

C des ends to a urve C/k su h that JacC ≃ A and so C is optimal. On the ontrary, if theobstru tion is not trivial, then C des ends to a urve C/k su h that Jac C is isomorphi tothe (unique) quadrati twist of A and so its Weil polynomial is (X2−mX + q)g. In parti ular#C(k) = q+1−gm and C is not optimal (and a tually C has the minimum number of pointsa genus g urve over k an have).4.1. The end of the genus 1 and 2 ases. Sin e a genus 1 urve over a nite eldalways has a rational point, it is an ellipti urve and Proposition 2.5 tells us when an optimalgenus 1 urve exists (for the value of Nq(1) see [Deu41 or [Ser83a).When g = 2, all genus 2 urves are hyperellipti so the obstru tion is always trivial and theresult is similar to Theorem 3.9, namely Publi ations mathématiques de Besançon - 2011

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108 Optimal urves of genus 1, 2 and 3Theorem 4.5 (Serre). There is no optimal urve of genus 2 over Fq if and only if q = 4or 9 or m2 − 4q ∈ −3,−4,−7.In [Ser83a, a losed formula for the value of Nq(2) is given. More re ently, ompleting thework started by many authors, we obtained in [HNR09 the omplete pi ture for abeliansurfa es, i.e. we determined whi h isogeny lasses ontain the Ja obian of a genus 2 urvesin terms of the oe ients of the Weil polynomial.5. The genus 3 aseAs there exist non hyperellipti genus 3 urves (the non-singular plane quarti s), Serre'sobstru tion may not be trivial. One hope is that, for ea h q, there would be an optimalhyperellipti urve but this possibility has to be dis arded: for instan e, there does not existany optimal hyperellipti genus 3 urves over F2n with n even sin e supersingular hyperellipti urves do not exists in hara teristi 2 [Oor91. Other ounterexamples an be found in odd hara teristi as well as it will be apparent in Proposition 5.6. Therefore, it is important tobe able to ompute Serre's obstru tion. Currently, there is no perfe t solution to this problembut we will summarize some of the ideas and partial answers whi h have been obtained.5.1. Spe ial families. The key-idea is to use some families of urves with non trivialautomorphisms, su h that their Ja obian is a produ t of ellipti urves expli itly obtained asquotient by ertain automorphism subgroups. Then one tries to reverse the pro ess and see ifone an glue given ellipti urves together to get a urve in the family. The possible quadrati extension one has to make during the onstru tion is Serre's obstru tion. Let us illustratethis pro edure with an example.Example 5.1. The following family represents genus 3 non hyperellipti urves in har-a teristi 2 with automorphism group ontaining (Z/2Z)2

C : (a(x2 + y2) + cz2 + xy + ez(x + y))2 = xyz(x + y + z), ac(a + c + e) 6= 0.The involutions are (x : y : z) maps to (y : x : z), (x + z : y + z : z) or (y + z : x + z : z). Toget the equation of the urve E1 = C/〈(x : y : z) 7→ (y : x : z)〉, one introdu es the invariantfun tions X = x + y, Y = xy and ndsE1 : (aX2 + c + Y + eX)2 = Y (X + 1).Doing similarly with the other involutions and rewriting the equations of the ellipti urves(see [NR10 for details) one gets that Jac C ∼ E1 × E2 × E3 where

E1 : y2 + xy = x3 + ex2 + a2(a + c + e)2,

E2 : y2 + xy = x3 + ex2 + c2(a + c + e)2,

E3 : y2 + xy = x3 + ex2 + c2a2.Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 109Conversely, we now want to glue ordinary ellipti urves Ei with j-invariant ji 6= 0. They analways be written Ei : y2 + xy = x3 + ex2 + 1/ji where, if q > 2, TrFq/F2(e) = 0 if and only if

Tr(Ei) ≡ 1 (mod 4). Let s4i = 1/ji, then

a =s1s3

s2,

c =s2s3

s1,

e =s1s3

s2+

s2s3

s1+

s1s2

s3.(1)Now, for instan e, assume that m ≡ −1 (mod 8). This happens for n = 35, 37, 63, . . .. We hoose E = E1 = E2 = E3 an ordinary ellipti urve with tra e −m and j-invariant j (Eexists sin e 2 ∤ m). Sin e we an assume that q > 4, the urve E has an 8-torsion point andit is not di ult to he k that this implies (a tually is equivalent to) TrFq/F2

1/j = 0. Hen eTrFq/F2

(

s1s3

s2+

s2s3

s1+

s1s2

s3

)

= TrFq/F2(1/j) = 0.On the other hand, sin e Tr(E) ≡ 1 (mod 4), we have TrFq/F2(e) = 0 as well, so there is noobstru tion to (1). A tually, we get an expli it equation

C : (j−1/4(x2 + y2 + z2 + xz + yz) + xy)2 = xyz(x + y + z)for the optimal urve.Exploiting other families of urves in hara teristi 2, we get the following result.Theorem 5.2 ([NR08, NR10). If n is even, there exists an optimal urve over F2n ifand only if n ≥ 6.If n is odd and m ≡ 1, 5, 7 (mod 8), there is an optimal urve over F2n .When n > 1 is odd and m is even, there is of ourse no optimal urve sin e there is no ellipti urve with tra e −m. So only the ase m ≡ 3 (mod 8) is missing to get a omplete answerwhen p = 2.More re ently, Mestre [Mes10 has worked with a family of urves with automorphism groupS3 and showed that if p = 3 (resp. p = 7), 3 ∤ m (resp. 3|m) and −m is a non-zero squaremodulo 7 (resp. n ≥ 7), then there exists an optimal urve over F3n (resp. F7n).To on lude on this approa h, let us point out that one ould use the family with auto-morphism group (Z/2Z)2 ( alled Ciani quarti s) also in hara teristi greater than 2, sin eSerre's obstru tion has been worked out in [HLP00. Unfortunately, one does not see whenthis obstru tion is trivial knowing only m (one needs the equations of the ellipti fa tors tode ide).5.2. Serre's analyti strategy. Inspired by results of Klein [Kle90, Eq.118,p.462 andIgusa [Igu67, Lem.10,11, in a 2003 letter to Jaap Top [LR08, Serre stated a strategy to ompute the obstru tion when the hara teristi is dierent from 2. Roughly speaking, hisidea was that a ertain Siegel modular form evaluated at a `moduli point' (A, a)/K is a squarePubli ations mathématiques de Besançon - 2011

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110 Optimal urves of genus 1, 2 and 3in K if and only if the obstru tion is trivial. In a series of three papers, it was shown thatthis is a urate (rst for Ciani quarti s, then in general) and how to ompute the obstru tionin the ase of the power of a CM ellipti urve. Let us state the general result without any omments on the proof whi h would lead us to far from our initial purpose (see however[Rit09, Chap.4 for details).Theorem 5.3 ([LRZ10). Let A = (A, a)/K be a prin ipally polarized abelian threefolddened over a eld K with char K 6= 2. Assume that a is geometri ally inde omposable. Thereexists a unique primitive geometri Siegel modular form of weight 18 dened over Z, denotedχ18, su h thati) (A, a) is a hyperellipti Ja obian if and only if χ18(A, a) = 0.ii) (A, a) is a non hyperellipti Ja obian if and only if χ18(A, a) is a non-zero square.Moreover, if K ⊂ C, let (ω1, ω2, ω3) be a basis of regular dierentials on A; γ1, . . . γ6 be a symple ti basis (for a) of H1(A, Z); Ωa := [Ω1 Ω2] = [

γjωi] be a period matrix with τa := Ω−1

2 Ω1 ∈ H3 a Riemann matrix.Then (A, a) is a Ja obian if and only if(2) χ18((A, a), ω1 ∧ ω2 ∧ ω3) :=(2π)54

228·∏

[ε] θ[ε](τa)

det(Ω2)18is a square in K.Let us re all that the Thetanullwerte θ[ε](τ) are the 36 onstants su h that[ε] =

[

ǫ1

ǫ2

]

∈ 0, 13 ⊕ 0, 13,with ǫ1tǫ2 ≡ 0 (mod 2) and for τ ∈ H3

θ

[

ǫ1

ǫ2

]

(τ) =∑

n∈Z3

exp(iπ(n + ǫ1/2)τt(n + ǫ1/2) + iπ(n + ǫ1/2)

tǫ2).Remark 5.4. For a dierent approa h on this result, see [Mea08.The initial aim of Serre's letter was of ourse the existen e of optimal urves of genus 3.However, one does not know how to ompute dire tly the value of χ18 over nite elds.Therefore, as Serre suggested, when A is ordinary, we lift (A, a) anoni ally over a numbereld and there, we use formula (2). Doing the omputation with enough pre ision, we anre ognize this value as an algebrai number. Finally we redu e it to the initial nite eld tosee if it is a square.As the Ja obian of an optimal urve is isogenous to the power of an ellipti urve E, in[Rit10, we worked out this pro edure expli it in the parti ular ase A = E3. Let a0 be theprodu t prin ipal polarization on E3 and M = a−10 a ∈ M3(End(E)). When End(E) is anorder in an imaginary quadrati eld, it is well known that M is the matrix of a prin ipalPubli ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 111polarization on E3 if and only if M is a positive denite hermitian matrix with determinant1 (see Remark 3.2 and [Mum08, p.209). Moreover, when E is dened over a number eld,we show how to translate the data (E3, a0M) into a period matrix of the orresponding torusin order to ompute the analyti expression of χ18. Let us illustrate this pro edure with thefollowing example.Example 5.5. Does there exist an optimal urve C of genus 3 over k = F47 ? If so,by Lemma 2.4 we know that Jac C is isogenous to E3 where E is an ellipti urve withtra e −⌊2

√47⌋ = −13. The urve E is then an ordinary ellipti urve and End(E) on-tains Z[π] ≃ Z[(13+

√132 − 4 · 47)/2] = Z[τ ] (where π is the k-Frobenius endomorphism and

τ = (1 +√−19)/2). Hen e End(E) = Z[π] is the ring of integers OL of L = Q(

√−19). Sin e

OL is prin ipal, E is unique up to isomorphism. Using the work of [S h98, one an see that,up to automorphism, there is a unique positive denite hermitian matrix M ∈ M3(OL) ofdeterminant 1 whi h is inde omposable. In the language of Se tion 3.1, this means that thereexists a unique positive denite unimodular inde omposable rank 3 hermitian OL-module.The abelian threefold (E3, a0M) is then the unique prin ipally polarized geometri ally inde- omposable abelian threefold with Weil polynomial (X2 + 13X + 47)3, up to isomorphism.Lifting E anoni ally over Q as E : y2 = x3 − 152x − 722 we an onsider the prin ipallyabelian threefold (E3, a0M) sin e End(E) = OL as well. Let [w1 w2] be a period matrix of Ewith respe t to the anoni al regular dierential dx/(2y). If we letΩ0 =

w1 0 0

0 w1 0

0 0 w1

w2 0 0

0 w2 0

0 0 w2

,

C3/Ω0Z6 ≃ E3(C) with the produ t polarization a0. We then need to nd a symple ti basisof Ω0Z6 for the polarization a0M . It is not di ult to prove that the rst Chern lass of a0Mwith respe t to the pull-ba k ωi of the dierentials dx/(2y) on ea h urve is represented bythe matrix

H =1

w1w2

tM.The alternated form T lassi ally asso iated to H on the latti e Ω0Z6 is T = Im(tΩ0HΩ0).One then nds a matrix B ∈ GL6(Z) su h that

BT tB =

[

0 I3

−I3 0

]and Ω = Ω0tB is a period matrix for the polarization a0M . Finally, one omputes an approx-imation of

χ = χ18((E3, a0M), ω1 ∧ ω2 ∧ ω3)thanks to the analyti formula (2) and we re ognize it as an element of L. We nd in our ase

χ = (219 · 197)2. Publi ations mathématiques de Besançon - 2011

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112 Optimal urves of genus 1, 2 and 3The value χ is a non-zero square over F47 so by Theorem 5.3 (ii) Serre's obstru tion is trivialand there is a non hyperellipti optimal urve of genus 3 over k.Similar omputations show that there is an optimal urve over Fq for q = 61, 137, 277 but notfor q = 311. Note that this result for q = 47 and q = 61 has already been obtained in [Top03using expli it models and the others have been onrmed by [AAMZ09. In [Rit10, tablesof values of χ as the one from Example 5.5 are given for (E3, a0M) where E is an ellipti urve with lass number 1 and M is taken from [S h98. From them, we an get for instan e:Proposition 5.6. Assume that q = pn is su h that 4q = m2 + d with d = 7 (resp. 19).Then there exists an (expli it) genus 3 optimal urve over Fq if and only ifm ≡ 1, 2 or 4 (mod 7) (resp. (m

19

)

(−2

p

)

= 1).Moreover if this urve exists, it is non hyperellipti .Assume that q = pn is su h that 4q = m2 + 43. In parti ular 43 is a square in Fp, let say43 = r2 with r ∈ Fp. Then there exists a genus 3 optimal urve over Fq if and only if

(m

43

)

(

α

p

)

= 1where α is either −2 · 3 · 7,−487,−47 · 79 · 107 · 173 or −15156± 8214r. Moreover if this urveexists, it is non hyperellipti .Remark 5.7. The term `expli it' in Proposition 5.6 omes from the fa t that for ertain(E3, a0M) of [Rit10 we were able to give the equation of a urve C su h that Jac C isisomorphi to (E3, a0M) using [Guà09. Hen e for these ases, we have a `universal' familyof expli it equations for the optimal urve.The fa t that the previous statement is embarrassedly umbersome reveals either the intrinsi di ulty of the problem or a wrong atta k angle. Moreover, the limits of this strategy alreadyappear in the example: the omputation of the anoni al lift, of the matri es M and of a periodmatrix make it algorithmi in nature. Worse, the omputation of an approximation of χ istime- onsuming sin e one has to re ognize it as an algebrai number (a tually for a good hoi e of the model E, χ is an algebrai integer). Therefore large values of the dis riminantof End(E) seem out of rea h.It might then be interesting to try to understand the prime de omposition of χ algebrai ally.Klein's formula linking χ18 to the square of the dis riminant of plane quarti s (see [Kle90,Eq.118,p.462 and [LRZ10, Th.2.23) makes us think about an analogue of the Néron-Ogg-Shafarevi h formula for ellipti urve [Sil92, Appendix C,16. We shall then interpret p|χ interms of the nature of (A, a) := (E3, a0M) (mod p). For instan e, using [I h95, p.1059, p|χif and only if (A, a) is geometri ally de omposable or a hyperellipti Ja obian. Unfortunately,we do not know how to dete t algebrai ally this last possibility (see the dis ussion in [Rit09,Se .4.5.1).Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 113Remark 5.8. We have not spoken yet about the ase q square when p > 2. First, whenp ≡ 3 (mod 4), one knows [Ibu93, p.2 that there exists an optimal genus 3 urve. This urve is even hyperellipti [Oor91 but not expli it (see however [KTW09 for some expli itsub- ases). Also Fermat urve x4 + y4 + z4 = 0 is optimal if n ≡ 2 (mod 4). Then, whenp ≡ 1 (mod 4) and n ≡ 2 (mod 4), Ibukiyama (lo . it.) shows that there is an optimal urve. Ibukiyama's strategy uses a mass formula on quaternion hermitian forms to show thedes ent of an inde omposable prin ipal polarization on a model over Fp of E3 where E/Fp2 isan ellipti urve with tra e −2p = −m. The abelian threefold and its quadrati twist beingisomorphi over Fp2 , he avoids the issue of omputing Serre's obstru tion.5.3. The geometri approa h. Following a onstru tion of Re illas [Re 74, we wereable to give in [BR10 a geometri hara terization of Serre's obstru tion. For the sake ofsimpli ity, let us assume that char k 6= 2 and that (A, a)/k is geometri ally the Ja obian of anon hyperellipti genus 3 urve. Sin e k is a nite eld, there exists a symmetri theta divisorΘ (for the polarization a) dened over k. Let Σ be the union of 2∗Θ and of the unique divisorin |2Θ| with multipli ity greater than or equal to 4 at 0.Proposition 5.9. Let α ∈ A(k) \ 0. The urve Xα = Θ ∩ (Θ + α) is smooth and onne ted if and only if α ∈ A(k) \ Σ.Hen e, the divisor Σ represents a bad lo us that needs to be avoided in the sequel. Assumingthat α /∈ Σ, the involution (z 7→ α− z) of Xα is xed point free and so Xα = Xα/(z 7→ α− z)is a smooth genus 4 urve.Proposition 5.10. The urve Xα is non hyperellipti and its anoni al model in P3 lieson a quadri Qα whi h is smooth.To go further, we need to assume that α is rational. When k is big enough, su h an α alwaysexists. We then obtain the following result.Theorem 5.11. Assume there exists α ∈ A(k) \Σ. Then (A, a) is a Ja obian if and onlyif δ = Disc Qα is a square in k∗.Let us sket h the proof. A non hyperellipti genus 4 urve X lies anoni ally in P3 on theinterse tion of a unique quadri Q and a ubi surfa e E. If we assume that Q is smooth,then X has two g1

3 oming from the two rulings of Q by interse ting them with E. Moreover,an easy omputation shows that Disc Q is a square if and only if these two g13 are denedover k. Now Re illas' onstru tion, whi h an be used when (A, a) is the Ja obian of a urve,shows that Xα has two (rather expli it) rational g1

3 . To on lude, it is then enough to showthat a quadrati twist of (A, a) (whi h is no more a Ja obian) leads to two onjugate g13 .The advantage of this approa h is that it stays over the nite eld k and is ompletelyalgebrai . Unfortunately, so far, we do not see how to ompute δ for A = E3 and a = a0MPubli ations mathématiques de Besançon - 2011

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114 Optimal urves of genus 1, 2 and 3in terms of an equation of E and the oe ients of M . The main di ulty seems to nd anequation (or even points) on a theta divisor in order to ompute an equation of Qα.Referen es[AAMZ09 E. Alekseenko, S. Aleshnikov, N. Markin, and A. Zaytsev. Optimal urves of genus 3 over nite elds with dis riminant -19, 2009. Available onhttp://www. itebase.org/abstra t?id=oai:arXiv.org:0902.1901.[BHV01 Yu. Bilu, G. Hanrot, and P. M. Voutier. Existen e of primitive divisors of Lu as and Lehmernumbers. J. Reine Angew. Math., 539:75122, 2001. With an appendix by M. Mignotte.[BR10 Arnaud Beauville and Christophe Ritzenthaler. Ja obians among abelian threefolds: a geo-metri approa h, 2010. to appear in Math. Annal.[Bro93 Bradley W. Bro k. Superspe ial urves of genera two and three. PhD thesis, Prin eton uni-versity, Prin eton, 1993.[Del69 Pierre Deligne. Variétés abéliennes ordinaires sur un orps ni. Invent. Math., 8:238243,1969.[Deu41 Max Deuring. Die Typen der Multiplikatorenringe elliptis her Funktionenkörper. Abh. Math.Sem. Hansis hen Univ., 14:197272, 1941.[Di 66 Leonard Eugene Di kson. History of the theory of numbers. Vol. II: Diophantine analysis.Chelsea Publishing Co., New York, 1966.[Eke87 Torsten Ekedahl. On supersingular urves and abelian varieties.Math. S and., 60(2):151178,1987.[Gop77 Valerii Denisovi h Goppa. Codes that are asso iated with divisors. Problemy Pereda£i Infor-ma ii, 13(1):3339, 1977.[Gop88 Valerii Denisovi h Goppa. Geometry and odes, volume 24 of Mathemati s and its Appli- ations (Soviet Series). Kluwer A ademi Publishers Group, Dordre ht, 1988. Translated from theRussian by N. G. Shartse.[Guà09 Jordi Guàrdia. On the torelli problem and ja obian nullwerte in genus three, 2009.http://arxiv.org/abs/0901.4376.[Hal10 Saa Haloui. The hara teristi polynomials of abelian varieties of dimensions 3 over niteelds. J. Number Theory, 130:27452752, 2010.[Has83 Ki-i hiro Hashimoto. Class numbers of positive denite ternary quaternion Hermitian forms.Pro . Japan A ad. Ser. A Math. S i., 59(10):490493, 1983.[Hay68 Tsuyoshi Hayashida. A lass number asso iated with the produ t of an ellipti urve withitself. J. Math. So . Japan, 20:2643, 1968.[HI83 Ki-i hiro Hashimoto and Tomoyoshi Ibukiyama. On lass numbers of positive denite binaryquaternion Hermitian forms. I,II,III. J. Fa . S i. Univ. Tokyo Se t. IA Math., 27,28,30:549601,695699,393401, 1980,1981,1983.[HK86 Ki-i hiro Hashimoto and Harutaka Koseki. Class numbers of positive denite binary andternary unimodular Hermitian forms. Pro . Japan A ad. Ser. A Math. S i., 62(8):323326, 1986.[HK89 Ki-i hiro Hashimoto and Harutaka Koseki. Class numbers of positive denite binary andternary unimodular Hermitian forms. Tohoku Math. J. (2), 41(2):171216, 1989.[HLP00 Everett W. Howe, Fran k Leprévost, and Bjorn Poonen. Large torsion subgroups of splitJa obians of urves of genus two or three. Forum Math., 12(3):315364, 2000.[HN65 Tsuyoshi Hayashida and Mieo Nishi. Existen e of urves of genus two on a produ t of twoellipti urves. J. Math. So . Japan, 17:116, 1965.Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 115[HNR09 Everett W. Howe, Enri Nart, and Christophe Ritzenthaler. Ja obians in isogeny lasses ofabelian surfa es over nite elds. Annales de l'institut Fourier, 59:239289, 2009.[Hof91 Detlev W. Homann. On positive denite Hermitian forms. Manus ripta Math., 71(4):399429, 1991.[Hon68 Taira Honda. Isogeny lasses of abelian varieties over nite elds. J. Math. So . Japan, 20:8395, 1968.[How95 Everett W. Howe. Prin ipally polarized ordinary abelian varieties over nite elds. Trans.Amer. Math. So ., 347(7):23612401, 1995.[How96 Everett W. Howe. Kernels of polarizations of abelian varieties over nite elds. J. Algebrai Geom., 5(3):583608, 1996.[How08 Everett W. Howe. Supersingular genus-2 urves over elds of hara teristi 3. In Computa-tional arithmeti geometry, volume 463 of Contemp. Math., pages 4969. Amer. Math. So ., Provi-den e, RI, 2008.[Ibu89 Tomoyoshi Ibukiyama. On automorphism groups of positive denite binary quaternion Her-mitian latti es and new mass formula. In Automorphi forms and geometry of arithmeti varieties,volume 15 of Adv. Stud. Pure Math., pages 301349. A ademi Press, Boston, MA, 1989.[Ibu93 Tomoyoshi Ibukiyama. On rational points of urves of genus 3 over nite elds. Tohoku Math.J. (2), 45(3):311329, 1993.[I h95 Takashi I hikawa. Tei hmüller modular forms of degree 3. Amer. J. Math., 117(4):10571061,1995.[Igu67 Jun-i hi Igusa. Modular forms and proje tive invariants. Amer. J. Math., 89:817855, 1967.[IKO86 Tomoyoshi Ibukiyama, Toshiyuki Katsura, and Frans Oort. Supersingular urves of genustwo and lass numbers. Compositio Math., 57(2):127152, 1986.[Kan97 Ernst Kani. The number of urves of genus two with ellipti dierentials. J. Reine Angew.Math., 485:93121, 1997.[Kle90 Felix Klein. Zur Theorie der Abels hen Funktionen. Math. Annalen, 36:388474, 1889-90.Gesammelte mathematis he Abhandlungen, XCVII, 388-474.[KO87 Toshiyuki Katsura and Frans Oort. Supersingular abelian varieties of dimension two or threeand lass numbers. In Algebrai geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages253281. North-Holland, Amsterdam, 1987.[KTW09 Tetsuo Kodama, Jaap Top, and Tadashi Washio. Maximal hyperellipti urves of genusthree. Finite Fields Appl., 15(3):392403, 2009.[Kuh88 Robert M. Kuhn. Curves of genus 2 with split Ja obian. Trans. Amer. Math. So ., 307(1):4149, 1988.[Lan06 Herbert Lange. Prin ipal polarizations on produ ts of ellipti urves. In The geometry ofRiemann surfa es and abelian varieties, volume 397 of Contemp. Math., pages 153162. Amer. Math.So ., Providen e, RI, 2006.[Lau02 Kristin Lauter. The maximum or minimum number of rational points on genus three urvesover nite elds. Compositio Math., 134(1):87111, 2002. With an appendix by Jean-Pierre Serre.[LR08 Gilles La haud and Christophe Ritzenthaler. On some questions of Serre on abelian threefolds.In Algebrai geometry and its appli ations, volume 5 of Ser. Number Theory Appl., pages 88115.World S i. Publ., Ha kensa k, NJ, 2008.[LRZ10 Gilles La haud, Christophe Ritzenthaler, and Alexey Zykin. Ja obians among abelian three-folds: a formula of Klein and a question of Serre. Math. Res. Lett., 17(2), 2010.[Mat58 Teruhisa Matsusaka. On a theorem of Torelli. Amer. J. Math., 80:784800, 1958.[Maz86 Barry Mazur. Arithmeti on urves. Bull. Amer. Math. So . (N.S.), 14(2):207259, 1986.Publi ations mathématiques de Besançon - 2011

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116 Optimal urves of genus 1, 2 and 3[Mea08 Stephen Meagher. Twists of genus 3 and their Ja obians. PhD thesis, RijksuniversiteitGroningen, 2008.[Mes10 Jean-François Mestre. Courbes de genre 3 ave S3 omme groupe d'automorphismes, 2010.http://arxiv.org/abs/1002.4751.[MN02 Daniel Maisner and Enri Nart. Abelian surfa es over nite elds as ja obians. ExperimentalMath., 11:321337, 2002. with an appendix of E.W. Howe.[MN07 Daniel Maisner and Enri Nart. Zeta fun tions of supersingular urves of genus 2. Canad. J.Math., 59(2):372392, 2007.[Mum08 David Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Resear hStudies in Mathemati s. Published for the Tata Institute of Fundamental Resear h, Bombay, 2008.With appendi es by C. P. Ramanujam and Yuri Manin, Corre ted reprint of the se ond (1974)edition.[MW71 James Stuart Milne and William C. Waterhouse. Abelian varieties over nite elds. 1969Number Theory Institute, Pro . Sympos. Pure Math. 20, 53-64, 1971.[NR08 Enri Nart and Christophe Ritzenthaler. Ja obians in isogeny lasses of supersingular abelianthreefolds in hara teristi 2. Finite elds and their appli ations, 14:676702, 2008.[NR10 Enri Nart and Christophe Ritzenthaler. Genus three urves with many involutions and appli- ation to maximal urves in hara teristi 2. In Pro eedings of AGCT-12, volume 521, pages 7185.Contemporary Mathemati s, 2010.[Oor75 Frans Oort. Whi h abelian surfa es are produ ts of ellipti urves? Math. Ann., 214:3547,1975.[Oor91 Frans Oort. Hyperellipti supersingular urves. In Arithmeti Algebrai Geometry (Texel,1989), Prog. Math., pages 247284, Boston, 1991. Birkäuser.[OU73 Frans Oort and Kenji Ueno. Prin ipally polarized abelian varieties of dimension two or threeare Ja obian varieties. J. Fa . S i. Univ. Tokyo Se t. IA Math., 20:377381, 1973.[Re 74 Sevin Re illas. Ja obians of urves with g1

4's are the Prym's of trigonal urves. Bol. So . Mat.Mexi ana (2), 19(1):913, 1974.[Rit09 Christophe Ritzenthaler. Aspe ts arithmétiques et algorithmiques des ourbes de genre 1, 2 et

3. Habilitation à Diriger des Re her hes, Université de la Méditerranée, 2009.[Rit10 Christophe Ritzenthaler. Expli it omputations of Serre's obstru tion for genus-3 urves andappli ation to optimal urves. LMS J. Comput. Math., 13:192207, 2010.[Ryb08 Sergey Rybakov. Zeta fun tions of algebrai surfa es and Ja obians of genus 3 urves overnite elds. PhD thesis, Mos ow State University, Me hani s andMath. Department, 2008. in russian,unpublished.[S h98 Alexander S hiemann. Classi ation of Hermitian forms with the neigh-bour method. J. Symboli Comput., 26(4):487508, 1998. tables available onhttp://www.math.uni-sb.de/ag/s hulze/Hermitian-latti es/.[Sek86 Tsutomu Sekigu hi. Erratum: On the elds of rationality for urves and for their Ja obianvarieties [Nagoya Math. J. 88 (1982), 197212; MR0683250 (85a:14021). Nagoya Math. J., 103:163,1986.[Ser68 Jean-Pierre Serre. Corps lo aux. Hermann, Paris, 1968. Deuxième édition, Publi ations del'Université de Nan ago, No. VIII.[Ser83a Jean-Pierre Serre. Nombres de points des ourbes algébriques sur Fq. In Seminar on numbertheory, 19821983 (Talen e, 1982/1983), pages Exp. No. 22, 8. Univ. Bordeaux I, Talen e, 1983.[Ser83b Jean-Pierre Serre. Sur le nombre des points rationnels d'une ourbe algébrique sur un orpsni. C. R. A ad. S i. Paris Sér. I Math., 296(9):397402, 1983.Publi ations mathématiques de Besançon - 2011

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Christophe Ritzenthaler 117[Ser85 Jean-Pierre Serre. Rational points on urves over nite elds, 1985. Le tures given at Harvard,notes by F.Q. Gouvéa.[Sha01 Vasily Shabat. Curves with many points. PhD thesis, Universiteit van Amsterdam, Amster-dam, 2001.[Shi79 Tetsuji Shioda. Supersingular K3 surfa es. In Algebrai geometry (Pro . Summer Meeting,Univ. Copenhagen, Copenhagen, 1978), volume 732 of Le ture Notes in Math., pages 564591.Springer, Berlin, 1979.[Sil92 Joseph H. Silverman. The arithmeti of ellipti urves, volume 106 of Graduate Texts in Math-emati s. Springer-Verlag, New York, 1992. Corre ted reprint of the 1986 original.[Tat66 John Tate. Endomorphisms of abelian varieties over nite elds. Invent. Math., 2:134144,1966.[Tat71 John Tate. Classes d'isogénie des variétés abéliennes sur un orps ni (d'après Honda). InSéminaire Bourbaki 1968/69, volume 179 of Le ture Notes in Math., pages 95110. Springer, Berlin,1971.[Top03 Jaap Top. Curves of genus 3 over small nite elds. Indag. Math. (N.S.), 14(2):275283, 2003.[Wat69 William C. Waterhouse. Abelian varieties over nite elds. Ann. S i. É ole Norm. Sup. (4),2:521560, 1969.[Wei48 André Weil. Variétés abéliennes et ourbes algébriques. A tualités S i. Ind., no. 1064 = Publ.Inst. Math. Univ. Strasbourg 8 (1946). Hermann & Cie., Paris, 1948.[WL01 Rui Qing Wang and Guo Sheng Li. Inde omposable denite Hermitian forms over imaginaryquadrati elds. J. Zhengzhou Univ. Nat. S i. Ed., 33(3):2227, 2001.[Xin96 Chaoping Xing. On supersingular abelian varieties of dimension two over nite elds. FiniteFields Appl., 2(4):407421, 1996.[Zhu97 Fu-Zu Zhu. On the onstru tion of inde omposable positive denite Hermitian forms overimaginary quadrati elds. J. Number Theory, 62(2):353367, 1997.12 o tobre 2010Christophe Ritzenthaler, Institut de Mathématiques de Luminy, UMR 6206 du CNRS, Luminy, Case907, 13288 Marseille, Fran e. • E-mail : ritzenthiml.univ-mrs.fr

Publi ations mathématiques de Besançon - 2011

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COMPUTING WITH HECKE GROSSENCHARACTERS

by

Mark Watkins

Abstract. — We describe how to compute with algebraic Hecke Grossencharacters. We firstdescribe Dirichlet characters on number field elements, and finite order Hecke characters onideals, before passing to the general case. Our primary interest is not directly in the charactersthemselves, but rather in their L-functions, and particularly the special values of these. Weavoid the adelic language, since it does not readily lend itself to computer implementation. Wegive many numerical examples that have been computed with the Magma computer algebrasystem.

Resume. — Nous montrons comment il est possible d’effectuer des calculs en utilisant les“Grossencharacters”. Dans un premier temps, on decrit les caracteres de Dirichlet des corpsde nombres ainsi que les caracteres de Hecke d’ordre fini sur les ideaux, avant de passer aucas general. Notre principal interet n’est pas directement les caracteres pour eux-memes, maisplutot les fonctions L associees, en particulier leurs valeurs speciales. Nous evitons le langageadelique puisqu’il n’est pas propre a l’implementation directe sur machine. Nous donnons denombreux exemples numeriques calcules avec le systeme de calcul formel Magma.

1. Introduction

We give a description of how to compute with algebraic Hecke Grossencharacters. Due to thecomputational nature of our undertaking, we try to avoid the adelic language as much as pos-sible, choosing a more explicit phrasing. We start by reviewing Dirichlet characters (on fieldelements) and finite order Hecke characters (on ideals), before passing to the Grossencharactercase. The algorithmic problems of interest include discrete logarithms for residue and rayclass groups (see [12]), and principalisation of ideals when Grossencharacters are introduced.We also need to be careful about embeddings and choices of extension fields at various places.Convenient references are Tate’s thesis [23] (though it uses the adelic language), and ChapterZero of Schappacher’s book [19].

Key words and phrases. — Hecke Grossencharacters, ray class groups, L-functions, special values.

This paper grew out of a talk given at a CIRM meeting in Luminy in Nov-Dec 2009 on Theorie des nombres

et applications, and the author thanks the organisers for the invitation. An implementation is contained inversion 2.16 of Magma [2]. Thanks also to N. P. Dummigan for sorting out some details in Example 6.1.

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120 Computing with Hecke Grossencharacters

Our end goal is not only to compute with the characters themselves, but also with theirL-functions. The work of Lavrik [14] generalises that of Hecke, and allows us to approximateany value of an L-function via a “rapidly converging” series, which in practise means that weneed to take about

√N terms of the L-series, where N is the conductor. We use the Magma

[2] implementation of Dokchitser [6]. As an example, computing the special values L(G, 2)in §6.1 to hundreds of digits takes only a few seconds.

1.1. Notation. — We let ζk denote an unspecified primitive kth root of unity, and write Na

for the norm of an ideal a. We will often write (say) an expression like p2 = (1 + i) = (e2),and take this to define e2 = 1 + i. A tensor product with no subscript will be taken to beover Q.

2. Dirichlet characters

Let K be a number field, and let I be an ideal contained in its ring of integers ZK . Reductionmodulo I yields a ring of residue classes corresponding to ZK/I, and the Dirichlet charactergroup modulo I is dual to the multiplicative group of units given by

(

ZK/I)⋆

. We candecompose this latter group over the prime power factors of I as

(

ZK/I)⋆ ∼=

pk‖I

(

ZK/pk)⋆,

which will allow us to restrict and induce characters.We let ΩR

K and ΩCK be respectively the sets of real and complex infinite places of K. We

can also include characters associated to ramification at real infinite places in ΩRK . Each real

infinite place ∞ splits the elements x ∈ K⋆ into two cosets depending upon the sign of theembedding x∞. We let χ∞(x) = sign(x∞) for elements x ∈ K⋆ and places ∞ ∈ ΩR

K , andnote that these χ∞ are multiplicative functions. By abuse of notation, for an ideal I andset Ω ⊆ ΩR

K , we write (ZK/IΩ)⋆ for the multiplicative group of invertible residue classes. Wethus have

(

ZK/IΩ)⋆ ∼=

pk‖I

(

ZK/pk)⋆ ×

∞∈Ω

(

ZK/∞)⋆,

and each of the terms in the latter product is isomorphic to Z/2.

2.1. Example. — We consider Dirichlet characters modulo I = (5) for the rationalfield K = Q. There are four of these, and since 2 is a primitive root mod 5, we can specifysuch a character via its value at 2, that is, χ(2) = ζi4 for some i. Two of these characters areeven (having χ(−1) = +1), and two of them are odd (with χ(−1) = −1).We can then consider the four additional characters that appear when we take Ω = ∞to consist of the infinite place. We now have a character χ∞(x) = sign(x), and its productwith each of the previous four characters yields four new ones. In particular, we obtain fourcharacters that are even, and thus trivial on the units, and two of these have ∞ in theirconductor, and two do not. Since these are trivial on the units, they will induce characterson ideals when we consider Hecke characters.

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Mark Watkins 121

2.2. Induction and restriction of characters. — In general, we can specify a Dirichletcharacter modulo IΩ by giving its image on generators for (ZK/IΩ)⋆. Thus it is easyto extend (or induce) a character χ to a larger modulus, as we simply note its values ongenerators for the new modulus.The operation of restriction is not quite so easy, but also has more importance as it allowsus to compute the conductor of a Dirichlet character. Given a Dirichlet character χ ofmodulus IΩ, we first decompose it into characters modulo prime powers pk‖I and placesin Ω. We restrict to a modulus pk by computing generators for (ZK/p

k)⋆ and requiringthat χp ≡ χ on these, while χp ≡ 1 on generators for

(

ZK/(IΩ/pk))⋆

, and together these giveus the desired behavior on IΩ via the Chinese remainder theorem. For infinite places ∞ ∈ Ω,it suffices simply to take χ∞ as above, though it is also convenient to determine generatorsfor (ZK/IΩc∞)⋆ where Ωc∞ = Ω\∞.Given a Dirichlet character χ of prime power modulus pk, we then want to compute thesmallest power pl for which it is a character (that is, trivial on 1 + plZK). We can assumethat k ≥ 2 and l ≥ 1, as else the problem is not difficult. We then iteratively let l decreasefrom k to 1, and at each step write down generators for the quotient of multiplicative groups(1+pl−1ZK)(1+plZK)

and determine if χ is trivial on them. This quotient is isomorphic to the additive

group of Fp∼= Ffp where Np = pf , and this simplifies the calculation. Upon finding the largest

l for which χ is nontrivial on this quotient, we conclude that the local conductor is pl.

2.2.1. Use of Dirichlet characters. — The value of Dirichlet characters is somewhat limitedas they do not have L-functions attached to them unless they are trivial on the units. Oneapplication of them would be to Hilbert modular forms with character.

3. Hecke characters

We now pass to Hecke characters, which are characters on ideals. In particular, a Dirichletcharacter only lifts to a Hecke character if it is trivial on all the units of K. We must alsoconsider characters corresponding to the class group ClK of K.The ray class group modulo I is defined as the quotient of the ideals of ZK that are coprimeto I by the principal ideals (α) for α ≡ 1 (mod I). We can further include information aboutramification at infinite places by requiring that such an α be positive at places in a set Ω.Given the standard multiplication operation on ideals, the ray class group is thus

RIΩ =a ⊆ ZK , gcd(a,I) = 1

(α) : α ≡ 1 (mod I), and α∞ > 0 for all ∞ ∈ Ω ,

and we have the following diagram:

(1)

1 −−−−→ (ZK/IΩ)⋆ −−−−→ AIΩ −−−−→ ClK −−−−→ 1∥

1 −−−−→ UK/IΩ −−−−→ AIΩ −−−−→ RIΩ −−−−→ 1

where AIΩ is a group extension of ClK by (ZK/IΩ)⋆ that we describe more fully below,and UK is the unit group of K.

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122 Computing with Hecke Grossencharacters

The Hecke character group modulo IΩ will be dual to RIΩ in the above, and as there is anatural quotient map from RIΩ onto ClK , there will be a natural injection from the Hilbert

character group ClK into the Hecke character group. Similarly, though not of as much import,there is an injection from UK into ZK that yields a quotient map from the Dirichlet character

group modulo IΩ onto the unit characters UK/IΩ.

3.1. The group extension AIΩ. — We now say a bit more about the group extension AIΩ.To construct this, we take representatives ai that generate the class group ClK and arecoprime to I. For each i we let oi be the order of ai in ClK , and we principalise a

oi

i = (ui)with (ui)∞ > 0 for all ∞ ∈ Ω and take a

oi

i = ui as a relation in AIΩ. We see that ui isdefined only up to units in general, but any ambiguity disappears when we take the quotientby UK . However, we still do have some choices with the various class group representatives.The computation of ray class groups is detailed in [12], and eventually reduces to computingdiscrete logarithms.

3.2. Dirichlet restriction and Hecke lifting. — We can restrict a Hecke character ψ toa Dirichlet character χ of the same modulus IΩ in an obvious way, simply by putting χ = ψon a set of generators for (ZK/IΩ)⋆. The reverse process lifts a Dirichlet character that istrivial on all units that are positive at all places in Ω, and yields a Hecke character that iswell-defined up to a Hilbert character.

3.3. Hecke character restriction and extension. — As with the Dirichlet charactercase, we often find it convenient to define a Hecke character ψ of modulus IΩ via giving valueson generators for RIΩ. This immediately gives us a method to restrict a Hecke character toa smaller modulus. When extending a Hecke character, the calculation of relative generatorsfor the Chinese remainder theorem can be a bit delicate, and it seems easier to work viaextending its Dirichlet restriction to the larger modulus as before and then combine this withevaluations of ψ on generators of RIΩ (see §3.5).

3.3.1. Lack of Hecke decompositions. — The conductor of a Hecke character is simply theconductor of its Dirichlet restriction, as the only information lost is that of a Hilbert character,whose conductor is trivial. There is really no precise idea of a “decomposition” for Heckecharacters, due to the possible interaction of units when restricting the modulus. An easyexample of this phenomenon is to take K = Q(

√37) and I = (3) and J = (5). Then RI has

order 1 and RJ has order 2, while RIJ is cyclic of order 4. Here we have that both RJ andRIJ are generated by (

√37). We see that the units −1, 6+

√37 generate the multiplicative

group (ZK/I)⋆, while they form subgroups of indices 2 and 4 respectively when consideredsimilarly modulo J and IJ .

3.4. Example. — Of the eight Dirichlet characters of Q modulo (5)∞ given in the previousexample, four of these are trivial on the units, and these lift to Hecke characters. Two ofthem have ∞ in their conductor.

3.5. Explicit lifting. — Take K = Q(√

229) which has class number 3 and I = (7).

Then we have that RI∼= Z/3 × Z/3, and we can identify one of these constituents with the

Hilbert characters. We can define ψ by ψ(p3) = ψ(2) = ζ3, though this still leaves ambiguity

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Mark Watkins 123

regarding p3 versus its conjugate, and similarly with ζ3. The Dirichlet group modulo I iscyclic of order 48, and ψ restricts to an element χ of order 3, which can be defined by χ(2) = ζ3and χ(2 +

√229) = 1.

We then extend χ to modulus IJ where J = p19 is a prime above 19. The Dirichlet grouphere is isomorphic to Z/48 × Z/18, where the element 2 has order 18 while 2 +

√229 has

order 144, and these generate (this is dependent on choosing J rather than its conjugate).

So we can define the lift of ψ to IJ by ψ(2) = ψ(p3) = ζ3 and ψ(2 +√

229) = 1.

3.6. L-function and functional equation. — Let ψ be a Hecke character of conduc-tor cΩ; we take ψ to be primitive so that the conductor and modulus are equal. The L-functionof ψ is given as an Euler product and Dirichlet series (convergent in some half-plane) as

L(ψ, s) =∏

p

(

1 − ψ(p)/Nps)−1

=∑

a⊆ZK

ψ(a)

Nas,

conventionally taking ψ(a) = 0 for ideals a that are not coprime to c.

We define ΓR(s) = Γ(s/2)

πs/2 and ΓC(s) = ΓR(s)ΓR(s+ 1) = 2 Γ(s)(2π)s , and write

L∞(ψ, s) =∏

∞∈ΩC

K

ΓC(s) ·∏

∞∈ΩRK

∞6∈Ω

ΓR(s) ·∏

∞∈ΩRK

∞∈Ω

ΓR(s + 1).

We then have that the completed L-function has a meromorphic continuation and satisfies

Λ(ψ, s) = ǫψΛ(ψ, 1 − s) for some ǫψ with |ǫψ| = 1, where

Λ(ψ, s) = L(ψ, s) ·(

Nc · |∆K |)s/2 · L∞(ψ, s).

We can also note we obtain the Dedekind ζ-function for K here by taking ψ to be the trivialHecke character – this is the only case where the L-function has a pole.

3.6.1. Root numbers. — We can quote the formula for root numbers from Tate’s thesis [23].The global root number ǫψ is a product

v ǫv(ψ) of local root numbers, where ǫv(ψ) = +1when v is finite and neither ψ nor K is ramified at it. We have three cases: v is archimedean,when ǫv(ψ) = e2πi/4 when v is real archimedean and v ∈ Ω, and else ǫv(ψ) = +1 (see [5,§5.3]); ψ is unramified at v finite, when ǫv(ψ) = ψ(dv)

−1 where dv is the local different of Kv;and ψ is ramified at v finite, when we get a Gauss sum for the local character ψv:

ǫv(ψ) =1√Ncv

a

ψv(a)e2πi·tr(a/πe

v),

where a runs over Zv/cv (or its multiplicative group), while πv is a uniformising elementand πev is a generator for the ideal dvcv, and finally the trace includes the canonical mapsQp → Qp/Zp → Q/Z → R/Z. Note that ǫψ is algebraic, though it is given by a specificembedding (given by the explicit e2πi in the above, rather than an arbitrary root of unity).In the case of Hilbert characters, we note that the above reduces to ǫψ = ψ(dK)−1, which leadsto various characterisations. For instance, when we have an integral power-basis ZK = Z[α],then the global different dK is principal (letting f be a minimal polynomial for α, we havethat dK =

(

f ′(α))

in this case), and so ǫψ = +1 for all Hilbert characters. The fact that theglobal different is always a square in the class group (see [11, Satz 176]) can also be exploited

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124 Computing with Hecke Grossencharacters

in some cases. An example of a Hilbert character with nontrivial root number is with thefield K = Q( 3

√175), where the class group has order 3 and the root numbers of the Hilbert

characters are the cube roots of unity. The different here is dK = p33p

25p

27, with p5 and p7 in

the same ideal class since p25p7 = ( 3

√175) is principal.

3.6.2. Critical values. — A critical point for a Hecke L-function is an integer n such thatneither L∞(s) nor L∞(1 − s) has a pole at s = n. The L-function of a (primitive) Heckecharacter ψ has critical points only when K is totally real and either all or none of the infiniteplaces are contained in Ω. Writing Q(ψ) for the value-field of ψ, a theorem of Siegel [22] (seealso [4, §1.2]) refines work of Klingen and states that L(ψ,−2k−ǫ) ∈ Q(ψ) for all nonnegativeintegers k, where ǫ = 1 when Ω is empty, and ǫ = 0 when Ω contains all the real places.

3.7. L-function example. — We take K = Q(√−23) of class number 3, and consider

either of the nontrivial Hilbert characters ψ. Thus any nonprincipal ideal gets mappedto ζ3 or ζ2

3 . Using a technique dating back to Hecke made explicit by Lavrik [14] we canwrite L(ψ, 1) as a rapidly converging series, and so approximate it as

L(ψ, 1) ≈ 0.368409320715826821111868466629.

Writing S for the (non-Galois) cubic subfield of the Hilbert class field of K, we can note that

ζS(s) = ζQ(s) · L(ψ, s),

and this allows us to use the residue formula for the Dedekind ζ-function to get that L(ψ, 1) =2π log ǫ√

23where ǫ is the real root of x3 − x − 1. We could alternatively write L(ψ, s) as the L-

function of a 2-dimensional Artin representation, but do not pursue such avenues herein.

3.8. An example with root numbers and critical values. — Let K = Q(√

13)and take ψ to be either nontrivial Hecke character modulo p13. At the bad prime p13

we have that dp13cp13 = (13) = (√

13)2, and so the root number ǫψ = ǫp13 is just given

by 1√13

j ψ(j)e2πi(2j/13) , where the sum is over residues mod 13. This is a root of x12 +113x

6 + 1, and an approximation is

0.711626069061866188779523 ± 0.7025584230735235114677424i.

We compute L(ψ,−1) = 413(7 − 11ζ3) and L(ψ,−3) = 4

13(3883 − 9491ζ3).Upon taking ψ to be either Hecke character with conductor p13∞1∞2, the global root numberis approximately

−0.8723658594356627130720628 ± 0.4888535642614028900637261i,

and at p13 we now get ǫp13 as a root of x6 − 113x

3 + 1, and we also have a contribution

of i2 = −1 from the two real infinite places. The first two critical values can be computed tobe L(ψ, 0) = 4

13(1 + 4ζ3) and L(ψ,−2) = 12(−4 + 5ζ3).

3.8.1. Relation of root numbers to other Dirichlet characters. — The reader can note thatwe get similar root numbers for the classical Dirichlet characters modulo 13. Indeed, we havetwo such characters of order 3, but the above root number formula would have e2πi(j/13),while in our case we have 2j in the exponent.

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Mark Watkins 125

3.9. PARI/GP implementation of Hecke characters. — There is a package of MaciejRadziejewski that implements Hecke characters in PARI/GP. See [15] for more details.

4. Hecke Grossencharacters

We largely follow Chapter Zero of [19], and we will only deal with algebraic Grossencharacters,or those of type A0. We let K be a number field, which we will eventually take to be a CM-field, that is, a totally imaginary quadratic extension of a totally real field.An ∞-type T is a sequence of integers (nσ) indexed by the embeddings σ ofK. The evaluationof an ∞-type at an element α ∈ K is given by T(α) =

σ(ασ)nσ . Note that the norm

is the ∞-type with all components equal to 1, and so we can easily renormalise to have(say) all the components nonnegative and at least one of them equal to zero. We requirethat nσ + nσ = w be constant over all embeddings σ, where here w is called the weight.For a given modulus IΩ, we consider the set of “coherent” ∞-types modulo IΩ on K thatis, the ∞-types for which T(α) = 1 for all units α ≡ 1 (mod I) with α∞ > 0 for all ∞ ∈ Ω.Every algebraic Hecke character factors through the relative norm from K to the maximalCM-subfield of K (see [19, §0.3]), and so unless K contains a CM-field, the only coherentinfinity types will be multiples of the norm. For simplicity we will assume that K is itselfCM from now on, and so Ω is empty (as there are no real places). We can also pair theembeddings in the ∞-type according to complex conjugacy.Given a modulus I and a coherent ∞-type T, we then define

(2) G(

(α))

= T(α) =∏

σ

(ασ)nσ

for all α ≡ 1 (mod I). This defines a Hecke Grossencharacter G up to a finite quotient thatis exactly the Hecke character group for I. Indeed, what we called Hecke characters are oftencalled “finite order Hecke characters” for this reason. We again conventionally take G(a) = 0

on the ideals a that are not coprime to I. We can also note that G(p) will be of size (Np)w/2

for p that are coprime to I.In general, the values of G will lie in an extension E/K given by roots of principalisations ofpowers of representatives of the class group. Furthermore, twisting by a (finite order) Heckecharacter mod I can enlarge the value-field by an additional cyclotomic factor.

4.1. Example. — The typical first examples here are for imaginary quadratic fields of classnumber one, and are related to elliptic curves with complex multiplication. For instance, upontaking K = Q(

√−1) and I = p3

2 where p2 is the ramified prime above 2, with T = [1, 0] forthe ∞-type we get a Hecke Grossencharacter whose L-series matches that of the congruentnumber curve. The condition α ≡ 1(mod I) can be described in terms of primary generatorsin this case.In this context, we can mention the “canonical” Grossencharacters of Rohrlich [17] for imag-inary quadratic fields, which are closely related to Hilbert characters.

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126 Computing with Hecke Grossencharacters

5. Implementation of Grossencharacters

We next turn to how compute with Grossencharacters, and this will actually help explainsome technical theoretical points. The implementation of Dirichlet and Hecke charactersneeds nothing more than to be able to compute generators for the residue and ray classgroups, and then to be able to compute discrete logs with them. For Grossencharacters wealso need to be able to principalise ideals. We also have a choice of how the extension field isembedded into C, though such embeddings are permuted by Hilbert characters (see below).

5.1. First method for Grossencharacters. — The most direct method to computeGrossencharacters modulo I for a coherent ∞-type T of a field K is to write down represen-tatives ai of a basis for the ray class group RI , and extend K by the roots T(ui)

1/oi whereoi is the order of ai in RI and (ui) = a

oi

i is a principalisation with ui ≡ 1 (mod I). Here

by T(ui)1/oi we mean a specific choice of root in an extension field.

We concentrate on the principal Grossencharacter G, as the others can be determined viathe finite quotient corresponding to RI . We take G(ai) = T(ui)

1/oi and can compute G onany ideal b with gcd(b,I) = 1 by using the rule

(3) G(b) = G(b ·∏

i

avi

i )/G(∏

i

avi

i ).

Here the bi are chosen to make b∏

i avi

i lie in the trivial class of RI , with the value of G onsuch classes then determined (after principalisation) by the ∞-type as in (2).We can note that the choices of ai do not make too much difference. Indeed, if a and b

are in the same class with ao = (ua) and bo = (ub), then K(

T(ua)1/o)

and K(

T(ub)1/o)

are isomorphic fields. This follows because we can write a/b = (t) with t ≡ 1 (mod I), sothat toξ = ua/ub with ξ ≡ 1(mod I) a unit, which yields T(ub) = toT(ξ)T(ua) where T(ξ) =1, implying the result.However, there are two problems with this method. The first is that the extension field canbe of quite large degree, and the second is that it is difficult to pass from one modulus toanother.

5.2. Second method for Grossencharacters. — A superior method is to realise theabove “translation” (3) in the class group of K rather than in the ray class group, thoughwe then need to take care with units.Given a CM-field K, we choose representatives ai of a basis of the class group. We write oifor the orders, and (ui) = a

oi

i for principalisations. Given an ∞-type T we then let χTUK

be a character on the units UK such that χTUK

(ǫ) = T(ǫ)−1 for all units ǫ ∈ K. We

define the value-field E of K(T) to be the compositum of the Ki = K(T(ui)1/oi), where we

make specific choices of roots. The value-field of a Grossencharacter G can firstly “twist”K(T) to use different choices of roots, and additionally can include a cyclotomic extensioncorresponding to a nontrivial choice of a finite order Hecke character.We now introduce the modulus I and let χ be a Dirichlet character modulo I that is a liftof χT

UK. We may try to take χ to be “quasi-minimal” in the sense that we define it as above

for generators of UK/I, and trivial on other factors in the (ZK/I)⋆ decomposition. Allpossible lifts will differ by a finite order Hecke character, as the quotient of any two lifts will

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Mark Watkins 127

be trivial on the units. We can still have multiple “quasi-minimal” choices for χ – one caseisK = Q(

√−15) with I = p3p5, where either quadratic character modulo I with χ(−1) = −1

suffices.It is convenient to choose an lift χ of χ to AI (recall diagram (1)), and we choose χ(ai, ui) =

χ(ui)1/oi for some choice of root of unity on the right. Since the group extension is by the

class group, it follows that Hilbert characters will permute these choices of roots.Given a character-lift χ, the resulting “principal” Grossencharacter G modulo I will giveG(ai) = T(ui)

1/oi χ(ai, ui) when evaluated at the class group representatives ai, while thevalue at a principal ideal (v) will be G

(

(v))

= T(v)χ(v). We can compute the value at anyother ideal as with the above translation (3). We have (v) = (ǫv) for any unit ǫ ∈ K, andcan check that we do indeed have

G(

(ǫv))

= T(ǫv)χ(ǫv) = T(ǫ)T(v)χ(ǫ)χ(v) = T(v)χ(v) = G(

(v))

.

We will also want our representatives ai to be coprime to a given ideal I, and will want tobe able to change our chosen representatives in some cases. As an example, we take K =Q(

√−23) and suppose we choose p2 as a class group representative for a calculation with

some G3 modulo (3), and p3 for a calculation with some G2 modulo (2). To work with G3G2

we want to convert both G2 and G3 to use a representative that is coprime to 6, say p13. Weachieve this simply via G3(p13) = G3(p13/p2)G3(p2), and similarly with 2 and 3 switched.We can then compute either G2 or G3 with p13 rather than the original choices of class grouprepresentatives.

5.3. L-function, functional equation. — Attached to a Grossencharacter G we thushave an ∞-type T, a modulus I, a finite order Hecke character ψ, and a Dirichlet character χ,though these last two can be combined into the lift χ if desired. The conductor of such aGrossencharacter is the conductor of the product of χ with the Dirichlet restriction of ψ. Thevalue-field K(G) is given by K(Tχ,ψ), where here K(Tχ) is the field generated by the G(ai).

The L-function of a Grossencharacter is defined as∏

p

(

1 − G(p)/Nps)−1

, and we note that

L(G, s) = L(G, s) when G(p) = G(p) for all primes p, as conjugate pairs appear togetherin the Euler product. The functional equation is of a similar form to that for finite orderHecke characters, though we now have the completed L-function satisfies Λ(G, s) = ǫG ·Λ(G, w+ 1− s). For the Γ-factor, we take each conjugate pair [p, q] in the ∞-type and writeit with p ≤ q (this is permissible via relabelling the embeddings in the conjugate pair), andeach then gives a factor of ΓC

(

s− p)

. Note that G has the conjugate ∞-type to that of G,so that it has the same Γ-factors. Assuming that G has been scaled to remove factors of thenorm, the completed Λ-function has a pole precisely when G is the trivial Grossencharacterof weight 0

5.3.1. Root numbers. — The root number formulæ of above also need to be modified slightly.At (complex) infinite places v of type [p, q] (again taking p ≤ q) we have ǫv(G) = iq−p

[5, §5.3], while at finite places v where K is ramified but G is not the local root number

is ǫv(G) = G(dv)/Ndv = G(dv). Finally, at finite places v where G is ramified the previousGauss sum now has ψv replaced by (ψχ)v . There are various useful formulæ relating twistsof L-series, of which we mention that if ψ is finite and gcd(cψ , cG) = 1, then

(4) ǫGψ = ǫG · ǫψ · ψ−1(cG) · G−1(cψ).

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128 Computing with Hecke Grossencharacters

5.3.2. Critical values. —The critical points (for the L-functions) of these Hecke Grossencharacters are those integers nsuch that neither L∞(s) nor L∞(w + 1 − s) has a pole at s = n. It follows that the criticalpoints for a Grossencharacter of ∞-type [0, w] are s = 1 . . . w, and in general they are s =(µ+1) . . . (w−µ) where µ is the largest p amongst the [p, q] conjugate pairs (again taking p ≤ qin each). For the right half of the critical points, that is, the critical u with u ≥ (w + 1)/2(for else the root number intervenes), a conjecture of Deligne [5, §2.8, §8] states that there isa period ω(G) ∈ C such that L(G, u)(2πi)w−u/ω(G) is in the value-field of G, and this is atheorem of Blasius [1] (refining work of Shimura [21], while Goldstein and Schappacher [7]handled the case where K is imaginary quadratic).A more complete statement here can be made by first defining ωT

K for a CM-field K and

∞-type T via periods of an abelian variety (of dimension 12 [K : Q]) that has CM by K and

is defined over the Hilbert class field H/K. This is not completely straightforward, as theperiods will naturally lie in H ⊗ C, and we want them in K(G) ⊗ C. After obtaining theperiod ωTK , we can then relate it to ω(G) for any G with ∞-type T. We go through this

process in some detail in Example 6.1 below, first following [7, §4] to get an explicit ω[2,0]K

in K = Q(√−23), and then relating it to a Grossencharacter with this ∞-type and trivial

conductor.In fact, since we have ωT1

K ωT2K = ωT1+T2

K , all the ωTK will be generated by those on a ba-

sis for the ∞-types, namely all ∞-types of weight 1 (that is, all conjugate pairs as [0, 1]or [1, 0]). This basis has size [K : Q], and can be halved upon considering complex conjuga-tion, in exact correspondence with the periods from the abelian variety. The multiplicationformula ω(G1G2) = ω(G1)ω(G2) is true up to a determinable factor in K(G) (see [19,II.1.8.1-3], with multiplicativity of the p, and their relation to the periods), while the effectof twisting by a finite order Hecke character can also be determined and adds at worst anabelian extension (see [19, II.3], and note that the last sentence in [9, §4] is incorrect, asper [19, II.3.4]).The net result is that L(G, u)(2πi)w−u/ω(G) ∈ K(G), where both L(G, u) and ω(G) areelements of K(G) ⊗ C that can be independently computed.

5.4. An example. — Let K = Q(√−23) and I = p23 be the ramified prime. We consider

the principal Grossencharacter G with ∞-type [1, 0], taking the Dirichlet character χ to bethe quadratic character modulo p23. The lift χ can be taken to be the trivial extension of χ,as the orders of the class group and unit group are coprime.We take p2 as our representative for the class group generator, and principalise p3

2 =

(3+√−23

2 ) = (u) so that E = K(u1/3). Letting p3 be in the same class as p2 in the classgroup, we compute that G(p3) is equal to

G(p3/p22)G(p2

2) = G

(

(

5 −√−23

8

)

)

u2/3 = −(

5 −√−23

8

)

u2/3,

the last step since χ(

5−√−23

8

)

= −1.

If we had chosen a different lift, say χ(u) = ζ3, then we would have G(p22) = u2/3ζ2

3 , whichcorresponds to a different embedding for E as an extension of K. We can also note that p2

is only defined up to complex conjugacy.

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Mark Watkins 129

5.5. The choice of embedding and Hilbert characters. — We next consider K =Q(

√−39) which has cyclic class group of 4, and let ψ be a Hilbert character of order 4

with ψ(p2) = ζ4. Let χ be either of the (imprimitive) quadratic Dirichlet characters withmodulus p3p13 and χ(−1) = −1, and G be the principal Grossencharacter with ∞-type [1, 0].

We take p2 as our representative of class group generator where p42 =

(

5+√−39

2

)

= (u). Wenote that χ(u) = 1 while χ(−u) = −1, and so take χ(p2, u) = 1 and χ(p2,−u) = ζ8 for someprimitive 8th root of unity.Note that twisting G by a Hilbert character merely changes the embedding. Indeed, if we

consider u1/4 as fixed and write Gj for the Grossencharacter for the embedding with ζj4u1/4

instead of u1/4, we get Gj(p2) = ζj4u1/4 = ψj(p2)u

1/4 = ψj(p2)G0(p2), and this same cal-culation passes over to all ideals (via multiplicativity, and triviality of Hilbert characters onprincipal ideals), and so we get Gj = ψj ·G0.

The computation for the choice of the lift χ is similar. Upon taking χ(p2, u) = ζj4 we get

Gχ(p2) = T(u)1/4χ(p2, u) = u1/4ζj4 = Gj(p2), with the computation again transferring to allideals via translation. The independence (up to embeddings) of the choice of a unit with u

can be verified by noting that the expressions T(u)1/4χ(p2, u) and T(−u)1/4χ(p2,−u) =

(−1)1/4T(u)1/4ζ8 differ by some 4th root of unity.We can also note here that while each of the four Hilbert characters has sign +1 in itsfunctional equation (since ψ(dK) = ψ(p3p13) = +1), two of the Gψj have sign +1 and twohave sign −1. Indeed, using the above formula (4) we have

ǫGψj = ǫG · ǫψj · G−1(1) · ψ−j(cG) = ǫG · ψ−j(cG),

and we note that ψ(cG) = −1 (where here cG is either p3 or p13, depending on the choiceof χ).

5.5.1. Caveat. — The above correspondence between Hilbert characters and embeddingsneeds to be adjusted slightly when T(ui) is a kth power for some k|oi (with k > 1). Forinstance, the principal [3, 0] Grossencharacter for K = Q(

√−23) has K as its value field, and

twisting by a nontrivial Hilbert character adjoins ζ3.

6. Assorted examples

6.1. Critical values and elliptic curves. —Let K = Q(

√−23) and G be a Grossencharacter of ∞-type [2, 0] for the trivial modulus.

There are three twists of G corresponding to Hilbert characters (and/or embeddings, asabove), and we let ψ be a nontrivial Hilbert character. There are a number of field extensionsthat arise here. We write E621 for the field defined by β3 − 6β− 3 = 0, so that E = KE621 isthe value-field of G (with a twist by Q(ζ3) when the embedding changes). For the real cubicsubfield H−23 of the Hilbert class field H/K we write α3 − α− 1 = 0.We obtain that the special values at the edge of the critical strip are, up to permutation,approximated as:

L(G, 2) ≈ 1.06110583266449728309907405960,

L(Gψ, 2) ≈ 1.23819100212426040400794384795.

L(Gψ2, 2) ≈ 0.670337208665839403747922477472,

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130 Computing with Hecke Grossencharacters

We can compute that

L(G, 2)L(Gψ, 2)L(Gψ2 , 2) =π3

233

1

3√

23

22∏

i=1

Γ(i/23)χ(i),

where χ is the quadratic Dirichlet character modulo 23 (see [19, III.1.4]), and can note that

this L-product is also equal to π2

5 L(G3, 4). For the root number, we compute G(p23) = −23

so that ǫp23(G) = −1; when combined with ǫ∞(G) = i2−0 = −1 this implies that ǫG = +1,as expected for self-dual G and even weight.

6.1.1. Ratios of the L-values. — The six pairwise ratios of the above L-values lie in thedegree 18 field E621 · Q(

√69) · H−23, and the cubes of these ratios are all conjugate in

the degree 6 field E621 · Q(√

69), which is the Galois closure of the cubic field E621, andis also the maximal real subfield of the Galois closure of the value-field. Indeed, the sixnontrivial L(Gψi, 2)3/L(Gψj , 2)3 ratios are given by the six embeddings of 1/541696 times

(23832√

69 + 182478)β2 − (79253√

69 + 221763)β + (−79355√

69 + 434509).

Letting t be this element, it has relative norm 1 in Q(√

69) we so we can apply Hilbert’sTheorem 90 [13, Satz 90, §54, Capitel XV, p. 272] to a 3-cycle σ of this S3-extension and get an

element v with σ(v)/v = t. We can further factor v = es for some e ∈ E621 and s ∈ Q(√

69),and take

e = (365β2 + 53β + 431) = −(5 + β − β2)9(2 − β2)3η21η

32

(unique up to Q-scaling), where the first two elements have respective norms 2 and 23, andthe other two are units η1 = −2 − β and η2 = 1 + 2β.The end result of this is that

(

L(Gι, 2)3)

ι≈ e⊗ 0.00059831771559950518493950 ≈ e⊗ z ∈ E621 ⊗ R,

where this indicates L(Gι, 2)3 ≈ ι(e)z for each embedding ι : E621 → R. I do not see how to

remove the cubing here, even if one passes to value-field E = KE621, as no rational multipleof e is a cube; nor do I see any redoing of the computation with different choices that wouldlead to e having a cube root. Indeed, the six nontrivial L(Gψi, 2)/L(Gψj , 2) ratios really dolive in the degree 18 field obtained from adjoining the real subfield H−23 of the Hilbert classfield to the degree 6 value-field.

6.1.2. Periods from an elliptic curve. — The period z of above corresponds to the periodof an elliptic curve C that is defined over the real subfield of the Hilbert class field ([21,

Corollary, Theorem 10]), and this curve can be obtained from the j-invariant j(

1+√−23

2

)

=

−53(2+α+α2)3(2+3α)3 as the curve must have complex multiplication by K (see [8, §12] formore on this). We need to enlarge the places of ramification to include those for K(j)/Q(j)(see also [18] in this regard), which turns out to be the prime above 23 that is not alreadyramified in Q(j). We can get the desired curve by starting with a curve with the givenj-invariant and then twisting away any ramification at undesired places.We obtain that C is given by

y2 + (1 + α)xy + y = x3 − x2 − (17 + 27α + 12α2)x− (45 + 72α + 61α2),

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Mark Watkins 131

and that the real period corresponding to the real embedding of H−23 is approximately givenby ω1 = 0.8174695113021061400156361160297 – as a check, the “imaginary” period differs bya factor of

√23 from this, as

(

1 +√−23

)

ω1/2. The other periods of C can be approximatedas

ω2 ≈ 1.84329374387271752388216654 ± 0.34697676998110995398120i,

with the lattices being given by[

ω2,1+

√−23

4 ω2

]

and[

ω2,−1+

√−23

4 ω2

]

, and these cor-

respond to the binary quadratic forms 2x2 ∓ xy + 3y2. Upon writing the productas Ω = ω1ω2ω2 ≈ 2.87595968338354139672, we can re-compute the above product ofL-values as

i L(Gψi, 2) ≈ 23216Ω2.

6.1.3. Relating the periods. — The above calculation computes the period directly for theGrossencharacter given by G NH/K : H → K and via [9, §4.7 (12)] (see also [19, II.1.8.11,

II.1.4.5]) we can relate it to ω(G3) via the formula

ω(G NH/K)

ω(G[H:K])=

1 1 1

α ασ (α)σ2

α2 (α2)σ (α2)σ2

= −√−23 ∈ K,

where we have used the fact that the factor that comes from the field discriminant (see [19,II.1.7.12(iv)]) is trivial here (as the field is Q). Since the value-field of G3 is K itself, bothperiods on the left are in K ⊗ C.We can also compute that L(G3, 4)(2πi)2 ≈ −5·23

54 Ω2, and in this manner can thus ob-

tain ω(G3) = (a + b√−23) ⊗ Ω2 ∈ K ⊗ C for some a, b ∈ Q as a direct experimental fact,

independent of the above relation to the Shimura periods. Furthermore, we can compute

that L(G3, 5)(2πi) ≈ 23√−23

216 Ω2, which shows how the extra factor of i in the 2πi interveneswhen raised to an odd power.

6.1.4. Reduction in periods. — We now proceed as in [7, §4.10-12, §9] to compute the periodfor G. We let σ ∈ Gal(H/K) be nontrivial, and take a = p2 as an element with nontrivialimage in the class group. This p2 gives an isogeny C → D for some elliptic curve D, and bytaking σ and p2 properly for Artin reciprocity, we have an isomorphism D → Cσ (where hereC,Cσ,D are taken to mean explicit models). We pullback a Neron differential from Cσ to Calong these maps, and get

Λ(p2) =1

2√−23

(

3 +√−23 + (1 −

√−23)α + (7 +

√−23)α2

)

in (4.10) of [7]. In (4.11), we thus get an element of relative norm 1 given by

Φ(σ) =1√−23

(

−8−3 +√−23

2β + 2β2

)

⊗ 1

4√−23

(

−9 + 3√−23 + (3 − 2

√−23)α+ (2 − 2

√−23)α2

)

as an element of E ⊗K H. The E-element is a cube root of u2 = −(3 +√−23)/2 and the

H-element h satisfies hσ(h)σ(σ(h)) = 1/u2, with p32 = (u2) in K.

As in (4.12) of [7] we then apply Hilbert’s Theorem 90 to find σ(x)/x = Φ(σ) with

23x = (17β2 − 30β − 22) ⊗ 1 + (−2β2 + 13β + 8) ⊗ α+ (9β2 − 24β − 13) ⊗ α2

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132 Computing with Hecke Grossencharacters

and this is in E621 ⊗H−23 (and unique up to elements in E621). This x corresponds to the[1, 0] ∞-type, and we will square it in the formulæ below to get the desired period for the

[2, 0] ∞-type. We can note(1) that x3 = (η2/η1)2 ⊗ α2.

We see that Theorem 9.1 of [7] says L(G, 2)/(xω)2 ∈ E ⊗ 1 upon choosing the same embed-ding H → C for both x ∈ E621 ⊗H → E621 ⊗ C and the elliptic curve C in the definitionof ω, and [7, §8] explains how to reduce the situation to real subfields.We embed H−23 → R with ω = ω1 and α ≈ 1.324717957244746025961, and so

(

L(Gι))

ι=

1

24 · 232(1767 + 165β − 307β2) ⊗ (xω1)

2 ∈ E621 ⊗R

when embedding both x and ω into the reals. Finally, we can note that

(

L(Gι)3)

ι=

1

29 · 3(365β2 + 53β + 431) ⊗ (α2ω31

)2 ∈ E621 ⊗ R

when embedding α into the reals, with an additional factor of −19 ± 4√−23 ∈ K on the

right when using the other embeddings. Indeed, we have that α2ω31 = Ω/3 under the real

embedding, with −2Ω/3 on the right with the others, which relates ω(G3) with ω(G)3.

6.2. A vanishing central symmetric-cube value. — This example is mentioned in [16,§4, Remark 2]. We take K = Q(

√−59), which has class number 3, and consider the prin-

cipal Grossencharacter G of ∞-type [3, 0] modulo p59 (with χ as the quadratic character).We see that G will take values in K, and indeed the L-function of G is the same as thatfor a rational weight 4 modular form of level 592. The interest of this example is that cen-tral value L(G, 2) = 0 vanishes even though the sign of the functional equation is even. Thisspecial value can be computed exactly as in [16], among other methods (such as modular sym-bols), so that the vanishing is known. To the best of my knowledge, the conjectural rephrasingas the leading term of the Taylor series L′′(G, 2) ≈ 5.752742016791747931531921931 as anexplicit regulator has not yet been achieved, though the works [20] and [3] do construct cyclesin Griffiths groups in similar contexts.We can note that [24, §6.6.2] gives more examples of such “analytic rank 2 motives” (inquadratic twist families corresponding to CM elliptic curves over Q), but also indicates (interalia) that there is no known degree 2 L-function of (motivic) weight greater than 1 whose“analytic rank” is more than 2, with a large amount of data computed in the Grossencharactercase.

6.3. An example over Q(ζ5). — The above examples have all dealt with imaginaryquadratic fields. We now give one over a larger field. In order to deal with ∞-types in a reason-able manner, we will fix embeddings. We first consider the ∞-type

(

[3, 0], [1, 2])

where the first

(1)This means that x has a square root in this field (note that we have chosen the E-scaling to ensure this, sothe content here is that the H-component is square), but I do not know if this is of import. It could be relatedto the fact that Shimura’s periods can also be used with “half-integral” Grossencharacters in some contexts(for instance, see Remark 1.7(a) of [10] – my understanding is that Shimura’s periods extend to the situationof an ∞-type such as [ 1

2,− 1

2], though I could be mistaken). Also, it becomes tricky to make refined statements

about integrality as the use of Hilbert’s theorem 90 gives us the freedom of an element of E = K(G), thoughsee [7, §10] for more in this regard.

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Mark Watkins 133

pair of (complex conjugate) embeddings sends ζ5 → e2πi(1/5), e2πi(4/5) and the second to ζ5 →e2πi(2/5), e2πi(3/5). We shall see that this is rather different from the

(

[3, 0], [2, 1])

∞-type.

We note T(ζ5) = (e2πi(1/5))3(e2πi(4/5))0(e2πi(2/5))1(e2πi(3/5))2 = (e2πi/5)11, and so any modu-lus I must afford a Dirichlet character of order 5 for the character χT

UKto lift. As above,

we are most interested in characters whose ramification is non-disjoint from that of thefield, and so we work with I = p2

5 where p5 is the totally ramified prime of norm 5. We

note that ǫ = 1 + ζ25 is a unit with T(ǫ) = −e2πi/5, and so take χ of order 10 such

that χ(ζ5) = (e2πi/5)−1 and χ(ǫ) = −(e2πi/5)−1. Since the class group is trivial, we cancompute the principal Grossencharacter G on any ideal directly via principalisation. Forinstance, we can write

(11)ZK =∏

σ

(1 − ζ5 − ζ35 )σ = (t1)(t4)(t2)(t3)

with χ(tj) = (e2πi/5)4j which leads to

G(t1) ≈ 8.2745751406263143974426646 + 35.53211795322830188628510i,

G(t3) ≈ 36.225424859373685602557335 + 4.326499018591243213111138i,

while G(t4) = G(t1) and G(t3) = G(t2).The functional equation in this case has Γ(s)Γ(s − 1) as the factor for L∞(G, s),with Λ(G, s) = ǫGΛ(G, 4 − s) where ǫG = ǫp5 = +1 can be computed via

4∑

j=1

4∑

k=0

χ(j + ke5)e2πi·tr((j+ke5)/e55) = 5, with (e5)

5 = (1 − ζ5)5 = p5

5 = dp5cp5 .

We compute that the central value is given by

L(G, 2) =1

57/2

Γ(1/5)3Γ(2/5)3

Γ(3/5)2Γ(4/5)2,

where these exponents could perhaps be derived directly from [19, II.4.1].If we now follow the same calculation with the

(

[3, 0], [2, 1])

∞-type, we find that T(ζ5) =

(e2πi/5)3·1+4·0+2·2+3·1 = 1.However we still have T(ǫ) = −1 and thus there is no Grossencharacter with trivial modulus.Instead we take I = p5 and χ of order 2 with χ(ǫ) = −1, and can compute that

L(G, 2) =π

515/4

Γ(1/5)7/2Γ(3/5)1/2

Γ(4/5)7/2Γ(2/5)1/2.

Again we have ǫp5 = +1 via the computation∑

j χ(j)e2πi·tr(j/e45) =

√5. The root-number

contribution from infinite places is again i3−0i2−1 = +1.

References

[1] D. Blasius, On the Critical Values of Hecke L-Series. Ann. Math. 124 (2), no. 1 (1986), 23–63.Available online from http://www.jstor.org/stable/1971386

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134 Computing with Hecke Grossencharacters

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[9] G. Harder, N. Schappacher, Special values of Hecke L-functions and Abelian integrals. In Arbeit-stagung Bonn 1984, Proceedings of the meeting held by the Max-Planck-Institut fur Mathematik,Bonn, June 15–22, 1984. Edited by F. Hirzebruch, J. Schwemer, S. Suter, Lecture Notes in Mathe-matics 1111 (1984), Springer-Verlag, 17–49.Available online from http://dx.doi.org/10.1007/BFb0084583

[10] M. Harris, L-functions of 2 × 2 Unitary Groups and Factorization of Periods of Hilbert ModularForms. J. Amer. Math. Soc. 6 (1993), no. 3, 637–719.Available online from http://www.jstor.org/stable/2152780

[11] E. Hecke, Vorlesungen uber die Theorie der algebraischen Zahlen. (German) [Lectures on theTheory of algebraic Numbers]. Akademische Verlagsgesellschaft, Leipzig, 1923.

[12] F. Hess, S. Pauli, M. E. Pohst, Computing the multiplicative group of residue class rings. Math.Comp. 72 (2003), no. 243, 1531–1548. See http://www.jstor.org/stable/4099848

[13] D. Hilbert, Die Theorie der algebraischen Zahlkorper. (German) [The Theory of the algebraicNumber fields]. Jahresbericht der Deutschen Mathematiker-Vereinigung [Yearly report of the GermanMathematical Society] 4 (1897), 175–546.Available online from http://resolver.sub.uni-goettingen.de/purl?GDZPPN002115344

[14] A. F. Lavrik, Functional and approximate functional equations for the Dirichlet function. Mat.Zametki, 3:5 (1968), 613–622.Online at http://mi.mathnet.ru/eng/mz6720

[15] M. Radziejewski, On the distribution of algebraic numbers with prescribed factorization properties,Acta Arith. 116 (2005), 153–171. See the author’s webpage at

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http://www.staff.amu.edu.pl/~maciejr/computations/

/multiplicities_WWW/ZetaChiZeros.html

[16] F. Rodriguez Villegas, Square root formulas for central values of Hecke L-series II. Duke Math.J. 72 (1993), no. 2, 431–440.Available online from http://projecteuclid.org/euclid.dmj/1077289426

[17] D. E. Rohrlich, On the L-functions of canonical Hecke characters of imaginary quadratic fields.Duke Math. J. 47 (1980), no. 3, 547–557.Available online from http://projecteuclid.org/euclid.dmj/1077314180

[18] D. E. Rohrlich, Elliptic curves with good reduction everywhere. J. London Math. Soc. (2) 25(1982), no. 2, 216–222.Available online from http://jlms.oxfordjournals.org/cgi/reprint/s2-25/2/216.pdf

[19] N. Schappacher, Periods of Hecke characters. Lecture Notes in Mathematics, 1301, Springer-Verlag, Berlin, 1988. xvi+160 pp.Available online from http://dx.doi.org/10.1007/BFb0082094

[20] C. Schoen, Complex multiplication cycles and a conjecture of Beilinson and Bloch. Trans. Amer.Math. Soc. 339 (1993), no. 1, 87–115.Available online from http://dx.doi.org/10.2307/2154210

[21] G. Shimura, On the zeta-function of an abelian variety with complex multiplication. Ann. of Math.(2) 94 (1971), 504–533. See http://www.jstor.org/stable/1970768

[22] C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen. (German) [Calculation ofZeta functions at integral Points]. Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. II 10 (1969),87–102.

[23] J. T. Tate, Fourier analysis in number fields and Hecke’s zeta-functions, Ph.D. Thesis, PrincetonUniv., Princeton, N.J., 1950, reprinted in Algebraic number theory: proceedings of an instructionalconference organized by the London Mathematical Society (a NATO Advanced Study Institute) withthe support of the International Mathematical Union, edited by J. W. S. Cassels and A. Frohlich,Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967, xviii+366 pp.

[24] M. Watkins, Some Heuristics about Elliptic Curves. Experiment. Math. 17 (2008), no. 1, 105–125.

22 mai 2010

Mark Watkins, Magma Computer Algebra Group, School of Mathematics and Statistics, Carslaw BuildingF07, University of Sydney, NSW 2006, AUSTRALIA • E-mail : [email protected]

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Comité de rédaction Directeur de la revue : Patrick Hild, directeur du laboratoire

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Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s

2011

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11S. Ballet et R. Rolland Families of curves over finite fields

P. Bruin Computing coefficients of modular forms

B. Conrey Applications of the asymptotic large sieve

P. Lebacque et A. Zykin Asymptotic methods in number theory and algebraic geometry

M. Mohyla et G. Wiese A computational study of the asymptotic behaviour of coefficient fields of modular forms

C. Ritzenhaler Optimal curves of genus 1, 2 and 3

M. Watkins Computing with Hecke Grössencharacters

Revue du Laboratoire de mathématiques de Besançon (CNRS UMR 6623)

Ce numéro de la revue est financé avec le concours de l’Agence nationale de la recherche (ANR) dans le cadre de son projet 07-BLAN-0248 « ALGOL ».

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