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Carrés symétriques

Carrés symétriques, formes modulairesarithmétiques et fonctions L p-adiques

Bertrand GORSSE, Fabienne JORY-HUGUE,Alexei PANTCHICHKINE, Juilien PUYDT, Gilles ROBERT

Institut Fourier, Université Grenoble-1

B.P.74, 38402 St.–Martin d’Hères, FRANCE

cette présentation est disponible à l’adresse :http://www-fourier.ujf-grenoble.fr/˜panchish/06if.pdf


Table des matières

Introduction

Formes modulaires arithmétiques et opérateurs différentiels

Familles de Coleman

Familles de carrés symétriques

Familles de produits triples

Énoncé du probleme pour les produits triples

Description de la méthode

Énoncé du résultat principal sur les produits triples

Distributions and admissible measures

Quelques avantages de la nouvelle méthode p-adique

2


Introduction

Familles p-adiques de formes modulaires

Le groupe de travail est centré sur l’article [PaTV], Two variablep-adic L functions attached to eigenfamilies of positive slope,contenant une solution du problème :Étant donnée une famille p-adique analytique

k 7→ fk =

∞∑

n=1

an(k)qn ∈ Q[[q]] d’une pente positive σ > 0,

construire une fonction L de deux variables, donnant uneinterpolation de valeurs spéciales L∗(fk , s, χ) sur tous le couples(s, k), avec 1 ≤ s ≤ k − 1

( voir [CoM], p.6)

3


Introduction

Formes modulaires et leurs fonctions LOn considèreles formes modulaires comme1) certaines séries de puissances de la variable q :

f =

∞∑

n=0

anqn ∈ C[[q]] et comme

2) certaines fonctions holomorphes

sur le demi-plan de Poincaré

H = {z ∈ C | Im z > 0}

où q = exp(2πiz),z ∈ H, et on considèreles fonctions L attachées

L(f , s, χ) =

∞∑

n=1

χ(n)ann−s

pour tout caractère de Dirichletχ : (Z/NZ)∗ → C∗.

Le corps de Tate Cp

On fixe un nombre premier p, et soit Cp = Qp

le corps de Tate(the completion of the fieldof p-adic numbers)

We fix an embeddingip : Q → Cp, and viewalgebraic numbers asp-adic numbers via ip.

4


Introduction

On étudie aussi les autres familles de fonctions L :les carrés symmétriques (travail de Berttrand GORSSE et de Gilles ROBERT),produits triples (travail de Siegfried BOECHERER et d’AlexeiPANTCHICHKINE) (voir aussi [PaB1]), sur lesfonctions L de formes modulaires de Siegel et de Hilbert-Siegel (travail deMichel COURTIEU, de JungJu CHOIE et d’Alexei PANTCHICHKINE).On essaye d’utiliser un cadre adélique pour la méthode de Rankin-Selberg pourune étude des valuers spéciales (travail de Fabienne JORY-HUGUE et de JulienPUYDT).Produits triples p-adiques des formes modulaires construits en collaborationavec S.BOECHERER (Mannheim, Allemagne), en 2004-2005, et nouvel article :“Admissible p-adic measures attached to triple products of elliptic cusp forms”soumis dans Documenta Math. en 2005 (volume spécial dédié à John COATES)ce travail est disponible à l’ adresse :http ://www-fourier.ujf-grenoble.fr/˜panchish/tc/tc.pdf.

5




Let A be a commutative ring (a subring of C or Cp)Arithmetical nearly holomorphic modular forms (in the sense ofShimura, [ShiAr] are certain formal series

g =

∞∑

n=0

a(n;R)qn ∈ A[[q]][R], with the property

that for A = C, z = x + iy ∈ H, R = (4πy)−1, the seriesconverges to a C

∞-modular form on H of a given weight k andDirichlet character ψ. The coefficients a(n;R) are polynomials inA[R]. If degR a(n;R) ≤ r for all n, we call g nearly holomorphic oftype r (it is annihilated by ( ∂

∂z)r+1, see [ShiAr]).

6



We use the notation Mk,r (N, ψ,A) or M(N, ψ,A) for A-modulesof such forms (In our constructions the weight k varies).A known example (see the introduction to [ShiAr]) is given by theseries

− 12R + E2 := −12R + 1 − 24∞∑

n=1

σ1(n)qn

=3π2

lims→0

y s∑

m1,m2∈Z

′(m1 + m2z)−2|m1 + m2z |

−2s , (R = (4πy)−1)

where σ1(n) =∑

d|n d .The action of the Shimura differential operator

δk : Mk,r (N, ψ,A) → Mk+2,r+1(N, ψ,A),

is given over C by δk(f ) = (1

2πi∂

∂z−

k4πy

)f .7


Familles de Coleman

Une famille de la pente σ > 0 de formes paraboliques fk depoids k ≥ 2 :

k 7→ fk =

∞∑

n=1

an(k)qn

∈ Q[[q]] ⊂ Cp[[q]]A model exampleof a p-adic family(not cusp and σ = 0) :Eisenstein series

an(k) =∑

d|n

dk−1, fk = Ek

1) the Fourier coefficients an(k) of fkand one of the Satake

p-parameters α(k) := α(1)p (k)

are given by certain p-adic analyticfunctions k 7→ an(k) for (n, p) = 1

2) the slope is constant and positive :ord(α(k)) = σ > 0

L’existence de familles d’une pente positive σ > 0 a été établi par Coleman, voir[CoPB]

R.Coleman gave an example withp = 7, f = ∆, k = 12a7 = τ(7) = −7 · 2392, σ = 1.

A program in PARI for computingsuch families is contained in [CST98](see also the Web-page of W.Stein,http ://modular.fas.harvard.edu/ )

8


Familles de Coleman

Théorie des modules de Banach-Coleman :• The operator U acts asa completely continuous operatoron each A-submodule M†(Npv ;A)⊂ A[[q]] (i.e. U is a limitof finite-dimensional operators)

=⇒ there existsthe Fredholm determinantPU(T )= det(Id − T · U) ∈ A[[T ]]

• there is a versionof the Riesz theory :for any inverse root α ∈ A∗

of PU(T ) there existsan eigenfunction g , Ug = αg

such that evk(g) ∈ Cp[[q]]are classical cusp eigenformsfor all k in a neigbourhoodB ⊂ X (see in [CoPB])

9


Familles de Coleman

Motivationcomes from the conjecture of Birch and Swinnerton-Dyer, see in [Colm03] , Colmez, P. :La conjecture de Birch et Swinnerton-Dyer p-adique. Séminaire Bourbaki. [ExposéN

◦.919] (Juin 2003). For a cusp eigenform f = f2, corresponding to an elliptic curve E byWiles [Wi95], we consider a family containing f .

One can try to approach k = 2, s = 1from the other direction, taking k → 2 ,instead of s → 1, this leads to a formulalinking the derivative over s at s = 1of the p-adic L-function with thederivative over k at k = 2of the p-adic analytic functionαp(k), see in [CST98] :

L′p,f (1) = Lp(f )Lp,f (1)

with Lp(f ) = −2dαp(k)

dk

∣∣k=2

0 1 2 3 4 50

1

2

3

4

5

s

k′

The validity of this formulaneeds the existence ofour two variable L-function !

10




Take Coleman’s family k ′ 7→ fk′ =

∞∑

n=1

an(k′)qn ∈ Q[[q]] of slope σ > 0 of cusp

eigenforms fk′ of weight k ′ ≥ 2 containing f , and consider the symmetric squareL-function :

D(s, fk′ , χ) = L(2s − 2k ′ + 2, ψ2χ2)∞∑

n=1

χ(n)an2(k ′)n−s = (4.1)

∏

l prime

{(1 − χ(l)α2

l (k′)l−s)(1 − χ(l)αl (k

′)βl (k′)l−s)(1 − χ(l)β2

l (k ′)l−s)}−1

◮ Holomorphy of the function : (4.1) G.Shimura, [Shi75]◮ Algebraicity for critical values of the function (4.1) Don Zagier, [Za77],

J.Sturm, [St80]◮ Admissible p-adic L-functions attached to (4.1) A.Dabrowski, D.Delbourgo,

[Da-De]

11



Question : To construct two variable p-adic symmetricsquares attached to Coleman’s families.

For ordinary families this was done by Hida, and for Coleman’sfamilies this is the topic of the PhD Thesis of B.Gorsse, (InstitutFourier, Grenoble).Il utilise les séries Eisenstein de Cohen-Zagier de poids demi-entier,et le critère général d’admissibilité du Théorème principal dans[PaJTNB], p. 816 avec κ = 2.See also [Go-Ro] for a related algebraic computation of a certainPetersson product.Related techniques were used by W.Kim (Berkeley) in [Kim], whodeveloped the method of Hida [Hi81], and suggested a conjecturaldescription of the zeroes of such L-function in terms of theramification points of the eigencurve.

12



Cas de produits triples (S.Boecherer, A.Pantchichkine)The triple product with a Dirichlet character χ is defined as the following complexL-function (an Euler product of degree eight) :

L(f1 ⊗ f2 ⊗ f3, s, χ) =∏

p∤N

L((f1 ⊗ f2 ⊗ f3)p , χ(p)p−s ), (5.2)

where L((f1 ⊗ f2 ⊗ f3)p,X )−1 = (5.3)

det

(18 − X

(α

(1)p,1

00α

(2)p,1

)⊗

(α

(1)p,2

00α

(2)p,2

)⊗

(α

(1)p,3

00α

(2)p,3

))

=∏

η

(1 − α(η(1))p,1 α

(η(2))p,2 α

(η(3))p,3 X )

= (1 − α(1)p,1α

(1)p,2α

(1)p,3X )(1 − α

(1)p,1α

(1)p,2α

(2)p,3X )·. . .·(1 − α

(2)p,1α

(2)p,2α

(2)p,3X ),

product taken over all 8 maps η : {1, 2, 3} → {1, 2}.

13



Critical values and functional equation

We use the corresponding normalized L function (see [De79], [Co],[Co-PeRi]), which has the form :

Λ(f1 ⊗ f2 ⊗ f3, s, χ) = (5.4)

ΓC(s)ΓC(s − k3 + 1)ΓC(s − k2 + 1)ΓC(s − k1 + 1)L(f1 ⊗ f2 ⊗ f3, s, χ),

where ΓC(s) = 2(2π)−sΓ(s).The Gamma-factor determines the critical valuess = k1, · · · , k2 + k3 − 2 of Λ(s), which we explicitely evaluate (like

in the classical formula ζ(2) =π2

6). A functional equation of Λ(s)

has the form :s 7→ k1 + k2 + k3 − 2 − s.

14


Énoncé du probleme pour les produits triples

Énoncé du probleme pour les produits triples : Given threep-adic analytic families fj of slope σj ≥ 0, to construct afour-variable p-adic L-function attached to Garrett’s tripleproduct of these families

(we show that this function interpolates the special values

(s, k1, k2, k2) 7−→ Λ(f1,k1 ⊗ f2,k2 ⊗ f3,k3 , s, χ)

at critical points s = k1, · · · , k2 + k3 − 2 for balanced weights k1 ≤ k2 + k3 − 2 ; weprove that these values are algebraic numbers afters dividing by certain “periods”).However the construction uses directly modular forms, and not the L-values inquestion, and a comparison of special values of two functions is done after theconstruction.

15


Description de la méthode

Pour les produits triples la méthode inclue :• a version of Garrett’s integral representation for the triple L-functions of the form :for r = 0, · · · , k2 + k3 − k1 − 2,Λ(f1,k1 ⊗ f2,k2 ⊗ f3,k3 , k2 + k3 − r , χ) =∫ ∫ ∫

(Γ0(N2p2v )\H)3

f1,k1(z1)f2,k2(z2)f3,k3(z3)E(z1, z2, z3;−r , χ)∏

j

(dxjdyj

y2j

)

where fj ,kj=: f 0

j ,kjis an eigenfunction of U∗

p in Mkj(Np, ψj ),

E(z1, z2, z3;−r , χ) ∈ MT (N2p2v )= Mk1(N

2p2v , ψ1) ⊗ Mk2(N2p2v , ψ2) ⊗ Mk3(N

2p2v , ψ3)is the triple modular form of triple weight (k1, k2, k3), and of fixed triple Nebentupuscharacter (ψ1, ψ2, ψ3), obtained from a nearly holomorphic Siegel-Eisenstein seriesFχ,r = G ⋆(z ,−r ; k , (Npv )2,ψ), of degree 3, of weight k = k2 + k3 − k1, and theNebentypus character ψ = χ2ψ1ψ2ψ3

16



Théorème principal sur les fonctions p-adiques analytiquesen quatre variables attachées aux produits triples)

1) The function Lf : (s, k1, k2, k3) 7→

⟨f0,E(−r , χ)

⟩

⟨f0, f0

⟩ depends

p-adic analytically on four variables(χ · y r

p , k1, k2, k3) ∈ X × B1 × B2 × B3 ;2) Comparison of complex and p-adic values : for all (k1, k2, k3) inan affinoid neighborhood B = B1 × B2 × B3 ⊂ X 3, satisfyingk1 ≤ k2 + k3 − 2 : the values at s = k2 + k3 − 2 − r coincide withthe normalized critical special values

L∗(f1,k1 ⊗ f2,k2 ⊗ f3,k3 , k2 + k3 − 2 − r , χ) (8.5)

(r = 0, · · · , k2 + k3 − k1 − 2),

for Dirichlet characters χ mod Npv , v ≥ 1, such that all threecorresponding Dirichlet characters χj have Np-completeconductors :

17



Main Theorem (continued)

χ1 mod Npv = χ, χ2 mod Npv = ψ2ψ3χ, (8.6)

χ3 mod Npv = ψ1ψ3χ,ψ = χ2ψ1ψ2ψ3.

The normalisation of L∗ in (8.5) is the same as in Theorem C below.3) Dependence on x ∈ X : let H = [2ordp(λ)] + 1. For any fixed(k1, k2, k3) ∈ B and x = χ · y r

p the function

x 7−→

⟨f0,E(−r , χ)

⟩

⟨f0, f0

⟩

extends to a p-adic analytic function of type o(logH(·)) of thevariable x ∈ X.

18



Outline of the proof

1) • At each classical weight (k1, k2, k3) let us use the equality

⟨f0,E(−r , χ)

⟩=⟨f0, πλ(E(−r , χ))

⟩

deduced from the definition of the projector πλ :Kerπλ :=

⋂n≥1 Im (UT − λI )n, Imπλ :=

⋃n≥1 Ker (UT − λI )n.

Notice that the coefficients of E(−r , χ) ∈ M(A) depend p-adicanalytically on (k1, k2, k3) ∈ B = B1 × B2 × B3, whereA = A(B1 × B2 × B3) is the p-adic Banach algebra ofrigid-analytic functions on B.

19



Interpolation to all p-adic weights :• At each classical weight (k1, k2, k3) the scalar product

⟨f0,E(−r , χ)

⟩is

given by the first coordinate of πλ(E(−r , χ)) with respect to an orthogonalbasis of M

λ(A) containing f0 with respect to Hida’s algebraic Petersson product

〈g , h〉a :=⟨gρ|(

0

Np−1

0

), h⟩, see [Hi90].

Let us extend the linear form ℓ(h) =〈f0,h〉〈f0,f0〉

(defined for classical weights), to

Coleman’s type submodule of overconvergent families h ∈ Mλ(A)† ⊂ M

λ(A) asthe first coordinate of h with respect to some A-basis of eigenfunctions of all(triple) Hecke operators Tq for q ∤ Np, having the first basis vectorf0 ∈ M

λ(A)†.The linear form ℓ can be characterized as a normalized eigenfunction of theadjoint Atkin’s operator, acting on the dual A-module of M

λ(A)† : ℓ(f0) = 1.In order to extend ℓ to h = E(−r , χ), we need to choose a certain representativeof E(−r , χ) in the A-submodule M

λ(A)† , which is locally free of finite rank.

20



A representative of E(−r , χ) in the (locally free of finiterank A-submodule) M

λ(A)†

Choose a (local) basis ℓ1, · · · , ℓn given by some triple Fouriercoefficients of the dual (locally free of finite rank) A-moduleM

λ(A)†∗.Then define

ℓ = β1ℓ1 + · · · + βnℓ

n,

where βi = ℓ(ℓi ) ∈ A, and ℓi denotes the dual basis of Mλ(A)† :

ℓj(ℓi ) = δij . At each p-adic weight (k1, k2, k3) ∈ B let us define

ℓ(E(−r , χ)) := β1ℓ1(E(−r , χ))+· · ·+βnℓ

n(E(−r , χ)) (belongs to A),

where βi = ℓ(ℓi ) ∈ A, and ℓi(E(−r , χ)) ∈ A are certain Fouriercoefficients of the seies E(−r , χ).

21



ConclusionThere exists an element

E(−r , χ) ∈ Mλ(A)† ⊂ M(A)†

such that ℓ(E(−r , χ)) = ℓ(E(−r , χ)) (at each weight (k1, k2, k3)). In

fact, let us define

E(−r , χ) := ℓ1(E(−r , χ))ℓ1 + · · · + ℓn(E(−r , χ))ℓn

⇒ ℓ(E(−r , χ)) = ℓ(ℓ1)ℓ1(E(−r , χ)) + · · · + ℓ(ℓn)ℓ

n(E(−r , χ))= β1ℓ

1(E(−r , χ)) + · · · + βnℓn(E(−r , χ))

= ℓ(E(−r , χ)) (at each weight (k1, k2, k3)).Thus, the dependence of ℓ(E(−r , χ)) ∈ A on (k1, k2, k3) ∈ X 3 is p-adicanalytic.

In order to prove the remaining statements 2), 3), the dependence onx = χ · y r

p is studied in the next section.22



Distributions and measures with values in Banach modules

A

VC(Y ,A)∪

Cloc−const(Y ,A)

(a p-adic Banach algebra)(an A-module)(the A-Banach algebraof continuous functions on Y )(the A-algebraof locally constant functions on Y )

23



Definition (Distributions and measures)

a) A distribution D on Y with values in V is an A-linear form

D : Cloc−const(Y ,A) → V , ϕ 7→ D(ϕ) =

∫

Y

ϕdD.

b) A measure µ on Y with values in V is a continuous A-linear form

µ : C(Y ,A) → V , ϕ 7→

∫

Y

ϕdµ.

The integral∫

Y

ϕdµ can be defined for any continuous function ϕ,

and any bounded distribution µ, using the Riemann sums.

24



Admissible measures of Amice-Vélu

Admissible measuresLet h be a positive integer. A more delicate notion of anh-admissible measure was introduced in [Am-V] by Y. Amice, J.Vélu (see also [MTT], [V]) :

Definition

a) For h ∈ N, h ≥ 1 let Ph(Y ,A) denote the A-module of locally

polynomial functions of degree < h of the variableyp : Y → Z×

p → A× ; in particular,

P1(Y ,A) = C

loc−const(Y ,A)

(the A-submodule of locally constant functions). Let also denoteCloc−an(Y ,A) the A-module of locally analytic functions, so that

P1(Y ,A) ⊂ P

h(Y ,A) ⊂ Cloc−an(Y ,A) ⊂ C(Y ,A).

25



Admissible measures of Amice-Vélu

Definition of admissible measures (continued)

b) Let V be a normed A-module with the norm | · |p,V . For a givenpositive integer h an h-admissible measure on Y with values in V isan A-module homomorphism

Φ : Ph(Y ,A) → V

such that for fixed a ∈ Y and for v → ∞ the following growthcondition is satisfied :∣∣∣∣∣

∫

a+(Npv )(yp − ap)h

′

dΦ

∣∣∣∣∣p,V

= o(p−v(h′−h)) (9.7)

for all h′ = 0, 1, . . . , h − 1, ap := yp(a)

The condition (9.7) allows to integrate the locally-analyticfunctions on Y along Φ using Taylor’s expansions ! This means :there exists a unique extension of Φ to Cloc−an(Y ,A) → V .26



La construction peut être fait en quelques étapesindépendants :

◮ 1) Construction of distributions Φj (on a profinite or adelic space Y likeY = A∗

K/K∗ for a number field K ) with values in an infinite dimensional

modular tower M(ψ) over complex numbers (or in an A-module of infinite rankover some algebra A).

◮ 2) Application of a canonical projector of type πα onto a finite dimensionalsubspace Mα(ψ) of Mα(ψ) (or over locally free A-module of finite rank oversome algebra A) :

πα(g) = (Uα)−vπα,0(Uv (g)) ∈ Mα(Γ0(Np), ψ,Cp) (this works only for

nonzero α !) (this is the α-characteristic projector of g ∈ M(Γ0(Npv+1), ψ,C)(independant of v)).

◮ 3) On démontre un critère général d’admissibilité (voir Théorème principal dans[PaJTNB], p. 816) disant que la suite πα(Φj) de distributions à valeurs dansMα(ψ) donne une h-mesure admissible Φ à valeur dans ce module de rank finipour un nombre naturel h convenable (déterminé par la pente ordp(α)).

27



◮ 4) Application of a linear form ℓ of typeg 7→ 〈f 0, πα(g)〉/〈f , f 〉 produces distributions µj = ℓ(πα(Φj )),and (automatically ) an admissible measure : the growthcondition is automatically satisfied starting from congruencesbetween modular forms πα(Φj )

◮ 5) One shows that certain integrals µj(χ) of the distributionsµj coincide with certain L-values ; however, these integrals arenot necessary for the construction of measures (already doneat stage 4).

◮ 6) One shows a resultat on uniqueness for the constructedh-admissibles measures : they are determined by many of theirintegrals over Dirichlet characters (not all), for example, onlyover Dirichlet characters with sufficiently large conductor (thisstage is not necessary, but it is nice to have uniqueness of inthe construction), see [JoH05].

28



◮ 7) If we are lucky, we can prove a functional equation for theconstructed measure µ (using the uniqueness in 6), and usinga functional equation for the L-values (over complex numbers,comuted at stage 5), for example, for Dirichlet characters withsufficiently large conductor (again, this stage is not necessary,but it is nice to have a functional equation)This strategy is applicable in various cases (described above),cf. [PaJTNB], [Puy], [Go02].

Note that the eigenspaces M(α) of U are contained in the primary

subspaces Mα, and they where used by D. Kazhdan, B. Mazur,

C.-G. Schmidt, see [KMS2000], in the p-ordinary case via a p–adiclimit procedure. Notice that we do not need a p–adic limitprocedure, and we treat the general case of any positive slope.

29


Références

Références

Amice, Y. and Vélu, J., Distributions p-adiques associées auxséries de Hecke, Journées Arithmétiques de Bordeaux (Conf.Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc. Math.France, Paris 1975, 119 - 131

Batut, C., Belabas, D., Bernardi, H., Cohen, H., Olivier, M. :The PARI/GP number theory system.http ://pari.math.u-bordeaux.fr

Böcherer, S., Über die Funktionalgleichung automorpherL–Funktionen zur Siegelscher Modulgruppe, J. reine angew.Math. 362 (1985) 146–168

Böcherer S., Über die Fourier–Jacobi Entwickelung SiegelscherEisensteinreihen. I.II., Math. Z. 183 (1983) 21-46 ; 189 (1985)81–100.

30


Références

Böcherer, S., Ein Rationalitätssatz für formale Heckereihen zurSiegelschen Modulgruppe. Abh. Math. Sem. Univ. Hamburg56, 35–47 (1986)

Boecherer, S., Heim, B. L-functions for GSp2 × Gl2 of mixedweights. Math. Z. 235, 11-51(2000)

Böcherer, S., Satoh, T., and Yamazaki, T., On the pullback ofa differential operator and its application to vector valuedEisenstem series, Comm. Math. Univ. S. Pauli, 41 (1992),1-22.

Böcherer, S., and Schmidt, C.-G., p-adic measures attached toSiegel modular forms, Ann. Inst. Fourier 50, N

◦5, 1375-1443(2000).

Böcherer, S. and Schulze-Pillot, R., On the central criticalvalue of the triple product L-function. David, Sinnou (ed.),Number theory. Séminaire de Théorie des Nombres de Paris

31


Références

1993–94. Cambridge : Cambridge University Press. Lond.Math. Soc. Lect. Note Ser. 235, 1-46 (1996).

Buecker, K., Congruences between Siegel modular forms on thelevel of group cohomology. Ann. Inst. Fourier (Grenoble) 46(1996), no. 4, 877–897.

Chandrasekharan, K. Arithmetical functions.Berlin–Heidelberg–New York, Springer–Verlag (1970).

Coates, J. On p–adic L–functions. Sem. Bourbaki, 40emeannee, 1987-88, n◦ 701, Asterisque (1989) 177–178.

Coates, J. and Perrin-Riou, B., On p-adic L-functions attachedto motives over Q, Advanced Studies in Pure Math. 17, 23–54(1989)

Coleman, Robert F., p-adic Banach spaces and families ofmodular forms, Invent. Math. 127, No.3, 417-479 (1997)

32


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