Triple products of Coleman’s families (a joint work in progress with S.Boecherer)

Triple products of Coleman’s families

Triple products of Coleman’s families (a joint work in progress with S.Boecherer)

A. A. Panchishkin

Institut Fourier, Université Grenoble-1

A talk for the French-German Seminar in Lille on November 9, 2005,

available at the address:

http://www-fourier.ujf-grenoble.fr/˜panchish/lille

Triple products of Coleman’s families

Contents

Introduction

Generalities on triple products Statement of the problem

Arithmetical nearly holomorphic modular forms Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura Statement of the Main Result

Distributions and admissible measures

Up–Operator and the method of canonical projection Critical values of theLfunctionL(f1⊗f2⊗f3,s, χ)(Theorem A) Theorems B-D

Scheme of the Proof Boecherer’s higher twist Ibukiyama’s differential operator Algebraic linear form

Evaluation ofp-adic integrals Criterion of admissibility

2

Triple products of Coleman’s families Introduction

Why study L -values attached to modular forms?

A popular proceedure in Number Theory is the following:

Construct a generating functionf =P_∞

n=0anqⁿ

∈C[[q]]of an arithmetical functionn7→an,

for examplean=p(n)

Computef via modular forms, for example X∞ n=0

p(n)qⁿ

= (∆/q)^−1/24

A number (solution)

Example 1[Chand70]:

(Hardy-Ramanujan) ↑ ↑

p(n) =e^π

√2/3(n−1/24) 4√

3λ²n +O(e^π√

2/3(n−1/24)/λ³_n), λn=p

n−1/24,

Good bases, finite dimensions, many relations and identities

Values ofL-functions, periods,

congruences, . . . Other examples: Birch and Swinnerton-Dyer conjecture,

. . .L-values attached to modular forms

3

Triple products of Coleman’s families Introduction

Our data: three primitive cusp eigenforms

fj(z) = X∞ n=1

a_n,jqⁿ∈S_k

j(Nj, ψj), (j =1,2,3) (1.1) of weightsk1,k2,k3, of conductorsN1,N2,N3, and of nebentypus charactersψjmodNj,N :=LCM(N1,N2,N3).

Letp be a prime,p∤N. We viewfj ∈Q[[q]]֒→^ip C_p[[q]]via a fixed embeddingQ֒→^ip Cp,Cp=Qb_p is Tate’s field.

Letχdenote avariableDirichlet character modNp^v,v ≥0.

We viewkj as avariableweight in the weight space

X =XNp^v =Homcont(Y,C^∗_p),Y = (Z/NZ)^∗×Z^∗_p∋(y0,yp).

The spaceX is ap-adic analytic space first used in Serre’s [Se73]

Formes modulaires et fonctions zêta p-adiques. Denote by (k, χ)∈X the homomorphism(y0,yp)7→χ(y0)χ(ypmodp^v)y_p^k. We write simplykj for the couple(kj, ψj)∈X.

4

Triple products of Coleman’s families Introduction

The purpose of this talk is to describe a four variable p -adic L function

attached to Garrett’s triple product of three Coleman’s families kj 7→

( fj,kj =

X∞

n=1

an,j(kj)qⁿ )

of cusp eigenforms of three constant slopes

σj =ordp(α⁽_p,j¹⁾(kj))≥0whereα⁽_p,j¹⁾(kj),α⁽_p,j²⁾(kj)are theSatake parametersgiven as inverse roots of the Heckep–polynomial 1−a_p,jX−ψj(p)p^kj−1X²= (1−α⁽_p,j¹⁾(p)X)(1−α⁽_p,j²⁾(p)X).

We assume thatord_p(α⁽_p,j¹⁾(kj))≤ord_p(α⁽_p,j²⁾(kj)).

This extends a previous result: (see [PaTV], Invent. Math. v. 154, N3 (2003)) where a two variablep-adicL-function was constructed interpolating on allk a function(k,s)7→L^∗(fk,s, χ)

(s=1,· · ·,k−1)for such a family.

5

Triple products of Coleman’s families Introduction

A family of slope σ > 0 of cusp eigenforms f

of weight k ≥ 2:

k7→fk = X∞

n=1

an(k)qⁿ

∈Q[[q]]⊂C_p[[q]]

A model example of ap-adic family (not cusp andσ=0):

Eisenstein series an(k) =X

d|n

d^k−1,fk =Ek

1) the Fourier coefficients an(k)offk

and one of the Satake p-parametersα(k) :=α⁽¹⁾p (k) are given by certainp-adic analytic

functionsk 7→an(k)for(n,p) =1 2) the slope is constant and positive:

ord(α(k)) =σ >0

The existence of families of slopeσ >0 was established in [CoPB]

R.Coleman gave an example with p=7,f = ∆,k=12

a7=τ(7) =−7·2392, σ=1.

A program in PARI for computing such families is contained in [CST98]

(see also the Web-page of W.Stein, http://modular.fas.harvard.edu/)

6

Triple products of Coleman’s families Introduction

Coleman proved:

•The operatorU acts as a completely continuous operator on eachA-submoduleM^†(Np^v;A)

⊂A[[q]](i.e. U is a limit of finite-dimensional operators)

=⇒there exists

theFredholm determinant P_U(T)

=det(Id−T·U)∈A[[T]]

•there is a version of theRiesz theory:

for any inverse rootα∈A^∗ ofPU(T)there exists an eigenfunctiong,Ug =αg

such thatevk(g)∈C_p[[q]]

are classical cusp eigenforms for allk in a neigbourhood B⊂X (see in [CoPB])

Triple products of Coleman’s families Generalities on triple products

The triple product with a Dirichlet characterχis defined as the following complexL-function (an Euler product of degree eight):

L(f1⊗f2⊗f3,s, χ) =Y

p∤N

L((f1⊗f2⊗f3)_p, χ(p)p^−s), (2.2)

whereL((f1⊗f2⊗f3)p,X)⁻¹= (2.3) det 18−X α⁽¹⁾_p,1

0 0 α⁽²⁾_p,1

!

⊗ α⁽¹⁾_p,2 0

0 α⁽²⁾_p,2

!

⊗ α⁽¹⁾_p,3 0

0 α⁽²⁾_p,3

!!

=Y

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 mapsη:{1,2,3} → {1,2}.

Triple products of Coleman’s families Generalities on triple products

Critical values and functional equation

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

Λ(f1⊗f2⊗f3,s, χ) = (2.4)

Γ_C(s)Γ_C(s−k3+1)Γ_C(s−k2+1)Γ_C(s−k1+1)L(f1⊗f2⊗f3,s, χ), whereΓ_C(s) =2(2π)^−sΓ(s).

The Gamma-factor determines thecritical values

s=k1,· · ·,k2+k3−2 ofΛ(s), which we explicitely evaluate (like in the classical formulaζ(2) =π²

6 ). A functional equationofΛ(s) has the form:

s7→k1+k2+k3−2−s.

9

Triple products of Coleman’s families Statement of the problem

Given three p -adic analytic families f

of slope σ

≥ 0, to construct a four-variable p -adic L -function attached to Garrett’s triple product of these families

(we show that this function interpolates the special values (s,k1,k2,k2)7−→Λ(f_1,k1⊗f_2,k2⊗f_3,k3,s, χ) at critical pointss=k1,· · ·,k2+k3−2 for balanced weights k1≤k2+k3−2; we prove that these values are algebraic numbers afters dividing by certain “periods”).

However the construction uses directly modular forms, and not the L-values in question, and a comparison of special values of two functions is doneafter the construction.

10

Triple products of Coleman’s families Statement of the problem

Our method includes:

•a version ofGarrett’s integral representationfor the triple L-functions of the form: forr =0,· · ·,k2+k3−k1−2, Λ(f_1,k1⊗f_2,k2⊗f_3,k3,k2+k3−r, χ) =

Z Z Z

(Γ0(N²p^2v)\H)³

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

j

(dxjdyj

y_j² ) where˜f_j,kj =:f_j,k⁰_j is an eigenfunction ofU_p^∗inM_k

j(Np, ψj), E(z1,z2,z3;−r, χ)∈M_T(N²p^2v)

=M_k

1(N²p^2v, ψ1)⊗M_k

2(N²p^2v, ψ2)⊗M_k

3(N²p^2v, ψ3)

is the triple modular form of triple weight(k1,k2,k3), and of fixed triple Nebentupus character(ψ1, ψ2, ψ3), obtained from a nearly holomorphic Siegel-Eisenstein series

Fχ,r =G^⋆(z,−r;k,(Np^v)²,ψ),of degree 3, of weight

k=k2+k3−k1, and the Nebentypus character ψ=χ²ψ1ψ2ψ₃

11

Triple products of Coleman’s families Statement of the problem

We obtain E ( z

, z

, z

; − r , χ) from a Siegel-Eisenstein series

by applying toFχ,r Boecherer’s higher twist(see (10.19)) andIbukiyama’s differential operator(see (10.20)).

These operations act explicitely on the Fourier expansions. Then one uses:

•Thetheory ofp-adic integrationwith values in Serre’s typeA-modules M_T(A)oftriple arithmetical nearly holomorphic modular formsoverp-adic Banach algebrasA. Explicit Fourier coefficientsaχ,r(R,T)∈Q[R,T]of E(−r, χ)are given by special polynomials of matriciesT= (tij),R= (Rij)and ofχ(β)β^r (withβ∈Z^∗_p∩Q) i.e. the coefficients ofaχ,r by some elementry p-adic measures

Z

Y

χy^rdµ^T∈A. HereA=A(B)is a certainp-adic Banach algebra offunctions on an open analytic subspaceB=B₁×B₂×B₃⊂X³in the product of three copies of the weight spaceX=Homcont(Y,C^∗_p).

These measures on the groupY = (Z/NZ)^∗×Z^∗_pproduce the coefficients of aχ,r ofE(−r, χ)ofM_T(A)for allp-adic weightsx∈X, given by

Z

Y

x(y)dµ^T∈A(an interpolation from x=χyp^r to all x∈X).

12

Triple products of Coleman’s families Statement of the problem

• The spectral theory of triple Atkin’s operator U = U

allows to evaluate the integral using at each weight(k1,k2,k3)the equalityD

f⁰,E(−r, χ)E

=D

f⁰, π_λ(E(−r, χ))E

with the projection π_λofM_T(A)to theλ-partM_T(A)^λ, defined by :

Kerπ_λ:=T

n≥1Im(UT−λI)ⁿ, Imπ_λ:=S

n≥1Ker(UT−λI)ⁿ. We prove thatU is a completely continuosA-linear operator on a certain Coleman’s submoduleM(A)^†of Serre’s type moduleM(A).

Then the projectionπ_λexists (on this submodule) due to general results of Serre and Coleman, see [CoPB], [SePB].

We show that there exists an element˜E(−r, χ)∈M(A)^† such that at each weight(k1,k2,k3)the equality holds:

Df⁰,E(−r, χ)E

=D

f⁰, π_λ(˜E(−r, χ))E

, and the product can be expressed through certain coefficients the seriesE˜(−r, χ)which are the same as those ofE(−r, χ).

13

Triple products of Coleman’s families Statement of the problem

• Key point: modular admissible measures

Let us write for simplicity: E(−r, χ)forE˜(−r, χ) M_T(A)instead of M_T(A)^†(Coleman’s submodule)

One definesadmissiblep-adic measuresΦ˜^λwith values in Banach A-modulesM^λ

T(A)which are locally free of finite rank, using the test functions:

Z

Y

χy_p^rΦ˜^λ=π_λ(E(−r, χ)).

•Passage from values in modular forms to scalar values: apply an algebraicA-linear formM^λ

T(A)→^ℓT Ato the constructed measure Φ˜^λ(in modular forms), and theevaluation mapsA^ev→^sCp for any p-adic triple weightss∈X³.

The linear formℓ_T is an algebraic version of the Petersson product (a geometric meaning ofℓ_T: the first coordinate in an (orthogonal) A-basis of eigenfunctions of all Hecke operatorsTq forq∤Np, with the first basis elementf₀∈M^λ(A)).

14

Triple products of Coleman’s families Statement of the problem

Using the evaluation map and the Mellin transform

We obtain themeasureµ=ℓ_T( ˜Φ^λ)with values inAon the profinite groupY.

•Construct an analytic functionL_µ:X →A=A(B)as the p-adic Mellin transformL_µ(x) =

Z

Y

x(y)dµ(y)∈A,x∈X.

•Solution: the function in questionL_µ(x,s)is given by evaluation ofL_µ(x)ats= (s1,s2,s3)∈B: this is ap-adic analytic function in four variables

(x,s)∈X×B₁×B₂×B₃⊂X×X×X×X

L_µ(x,s) :=evs(L_µ(x)) (x∈X, s∈B₁×B₂×B₃, L_µ(x)∈A).

Triple products of Coleman’s families Statement of the problem

Final step: comparison between C and C

•We check an equality relating the values of the constructed analytic functionL_µ(x,s)at the arithmetical characters

x=y_p^rχ∈X, and at triple weightss= (k1,k2,k3)∈B, with the normalized critical special values

L^∗(f1,k1⊗f2,k2⊗f3,k3,k2+k3−2−r, χ) (r =0,· · ·,k2+k3−k1−2), for certain Dirichlet charactersχmodNp^v,v≥1. These are algebraic numbers, embedded intoCp=Qb_p (the Tate field of p-adic numbers). The normalisation ofL^∗includes at the same time Gauss sums, Petersson scalar products, powers ofπ, the productλp(k1,k2,k3), and a certain finite Euler product.

Triple products of Coleman’s families

Arithmetical nearly holomorphic modular forms

Arithmetical nearly holomorphic modular forms (the elliptic case)

LetAbe acommutative ring(a subring of CorCp)

Arithmetical nearly holomorphic modular forms(in the sense of Shimura, [ShiAr] are certain formal series

g= X∞ n=0

a(n;R)qⁿ∈A[[q]][R], with the property that forA=C,z=x+iy ∈H, R= (4πy)⁻¹, the series converges to aC^∞-modular form onHof a given weightk and Dirichlet characterψ. The coefficientsa(n;R)are polynomials in A[R]. If deg_Ra(n;R)≤r for alln, we callg nearly holomorphic of typer (it is annihilated by(_∂z^∂)^r+1, see [ShiAr]).

17

Triple products of Coleman’s families Arithmetical nearly holomorphic modular forms

We use the notationM_k,r(N, ψ,A)orM˜(N, ψ,A) forA-modules of such forms (In our constructions the weightk varies).

A known example (see the introduction to [ShiAr]) is given by the series

−12R+E2:=−12R+1−24 X∞ n=1

σ1(n)qⁿ

= 3 π² lim

s→0y^s X

m1,m2∈Z

′(m1+m2z)⁻²|m1+m2z|^−2s,(R = (4πy)⁻¹)

whereσ1(n) =P

d|nd.

The action of theShimura differential operator δk :M_k,r(N, ψ,A)→M_k+2,r+1(N, ψ,A), is given overC byδk(f) = ( 1

2πi

∂

∂z − k 4πy)f.

18

Triple products of Coleman’s families

Arithmetical nearly holomorphic modular forms

Triple arithmetical modular forms

LetAbe a commutative ring. The tensor product overA M_k,r,T(N, ψ,A) :=M_k

1,r(N, ψ1,A)⊗M_k

2,r(N, ψ2,A)⊗M_k

3,r(N, ψ3,A) consists oftriple arithmetical modular formsas certain formal series of the form

g= X∞ n₁,n2,n3=0

a(n1,n2,n3;R1,R2,R3)qⁿ₁¹qⁿ₂²q₃ⁿ³

∈A[[q1,q2,q3]][R1,R2,R3], wherezj =xj+iyj ∈H, Rj = (4πyj)⁻¹, with the property that forA=C, the series converges to a

C^∞-modular form onH³of a given weight(k1,k2,k3)and character(ψ1, ψ2, ψ3),j=1,2,3. The coefficients

a(n1,n2,n3;R1,R2,R3)are polynomials inA[R1,R2,R3]. Examples of such modular forms come from the restriction to the diagonal of Siegel modular forms of degree 3.

19

Triple products of Coleman’s families Siegel-Eisenstein series

Siegel modular groups

LetJ2m=

0m −1m

1m 0m

.The symplectic group Sp_m(R) =

g ∈GL2m(R)|^tg·J2mg =J2m , acts on the Siegel upper half plane

Hm=

z=^tz∈Mm(C)|Imz >0

by g(z) = (az+b)(cz+d)⁻¹, where we use the bloc notation g =

a b c d

∈Sp_2m(R). We use thecongruence subgroup Γ^m₀(N) ={γ∈Sp_m(Z)|γ≡ ^∗0

∗

∗

modN} ⊂Sp_m(Z).

20

Triple products of Coleman’s families Siegel-Eisenstein series

A Siegel modular form

f ∈M_k(Γ^m₀(N), χ)of degreem>1, weightk and a Dirichlet chracterχmodNis a holomorphic functionf: H_m→Csuch that for everyγ=

a b c d

∈Γ^m₀(N)one has

f(γ(z)) =χ(detd)det(cz+d)^kf(z).

TheFourier expansionoff uses the symbolq^T=exp(2πitr(Tz))

=Qm i=1q_ii^Tii Q

i<jq_ij^2Tij ⊂C[[q11, . . . ,qmm]][qij, q⁻_ij¹]_{i,j=1,···,m},, qij=exp(2π(√

−1z_i,j)), andT in the semi-group Bm={T=^tT ≥0|T half-integral}:

f(z) = P

T∈Bm

a(T)q^T∈C[[q^Bm]](a formalq-expansion ∈C[[q^Bm]]),

21

Triple products of Coleman’s families Siegel-Eisenstein series

Siegel-Eisenstein series

Example ([Nag2], p.408)

E₄⁽²⁾(z) =1+240q11+240q22+2160q²11+ (240q⁻₁₂²+13440q⁻₁₂¹ 30240+13440q12+240q₁₂²)q11q22+2160q₂₂² +. . . E₆⁽²⁾(z) =1−504q11−504q22−16632q²11+ (−540q₁₂⁻²+44352q₁₂⁻¹

166320+44352q12−504q12²)q11q22−16632q22² +. . . .

22

Triple products of Coleman’s families Siegel-Eisenstein series

Arithmetical nearly holomorphic Siegel modular forms

Arithmetical Siegel modular forms

Consider a commutative ringA, the formal variables

q= (qi,j)i,j=1,...,m,R= (Ri,j)i,j=1,...,m, and the ring offormal Fourier series

A[[q^Bm]][R_i,j] =



f = X

T∈Bm

a(T,R)q^T

a(T,R)∈A[R_i,j]



 (5.5)

(over the complex numbers this notation corresponds to q^T=exp(2πitr(Tz)),R = (4πIm(z))⁻¹).

The formal Fourier expansion of a nearly holomorphic Siegel modular formf with coefficients inAis a certain element of A[[q^Bm]][R_i,j]. We callf arithmeticalin the sense of Shimura [ShiAr], ifA=Q.

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Maass differential operator

Let us consider theMaass differential operator(see [Maa])∆_mof degreem, acting on complexC^∞-functions onH_mby:

∆m=det( ˜∂ij), ∂˜ij=2⁻¹(1+δij)∂/∂ij, (5.6) its algebraic version is theRamanujan operator of degreem:

Θ_m:=det( 1

2πi∂˜_ij) =det(θij) = 1

(2πi)^m∆_m, (5.7) whereΘ_m(q^T) =det(T)q^T.

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Shimura differential operator

TheShimura differential operator(see [Shi76, ShiAr]):

δkf(z) =det(R)^k+1−κΘm

hdet(R)^κ−1−kfi

, whereR = (4πy)⁻¹, acts on arithmetic nearly holomorphic Siegel modular forms, and the composition is defined

δ_k^(r)=δ_k+2r−2◦ · · · ◦δ_k :Me^m

k(N, ψ;Q)→Me^m

k+2rm(N, ψ;Q), (5.8) where

δ_kf(z) = −1

4π m

det(y)⁻¹det(z−¯z)^κ−k∆m

hdet(z−z¯)^k−κ+1f(z)i .

25

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Universal polynomials Q ( R, T ; k, r )

Letf = X

T∈Bm

c(T)q^T∈M^m_k(N, ψ)be a formal holomorphic Fourier expansion. One shows thatδ^(r)_k f is given by

δ^(r)_k f = X

T∈Bm

Q(R,T;k,r)c(T)q^T.

26

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Universal polynomials (continued)

Here we use a universal polynomial (5.9) which can be defined for allk∈C, and it expresses the action of the Shimura operator on the exponential (of degreem):

δ_k^(r)(q^T) =Q(R,T;k,r)q^T. Ifm=1,r arbitrary (see [Shi76]),

δ_k^(r)= Xr

j=0

(−1)^r−j r

j

Γ(k+r) Γ(k+j)R^r−jθ^j, Q(R,n;k,r) =

Xr

j=0

(−1)^r−j r

j

Γ(k+r) Γ(k+j)R^r−jn^j.

27

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Universal polynomials (continued)

Ifr =1,marbitrary, one has (see [Maa]):

δ_kf(z) = X

T∈Bm

c(T) Xm

l=0

(−1)^m−lc_m−l(k+1−κ)tr ^tρ_m−l(R)·ρ^⋆_l(T) q^T whereR= (4πy)⁻¹= (R_i,j)∈Mm(R),cm(α) = Γm(α+κ)

Γ_m(α+κ−1), Γm(s) =π^m(m−1)/4

m−Y1 j=0

Γ(s−(j/2)).

Here we usethe natural representationρr :GLm(C)−→GL(∧^rC^m) (0≤r ≤m) of the group GLm(C)on the vector spaceΛ^rC^m. Thusρr(z)is a matrix of size ^m_r

× ^mr

composed of the subdeterminants ofz of degreer. Putρ^⋆_r(z) =det(z)ρ_m−r(^tz)⁻¹. Then the representationsρr andρ^⋆_r turn out to be polynomial representations.

28

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

In general (see [CourPa], Theorem 3.14) one has:

Q(R,T) =Q(R,T;k,r) (5.9)

= Xr t=0

r t

det(T)^r−t X

|L|≤mt−t

RL(κ−k−r)QL(R,T), QL(R,T) =tr ^tρ_m−l1(R)ρ^⋆_l1(T)

·. . .·tr ^tρ_m−lt(R)ρ^⋆_lt(T) ).

In (5.9),Lgoes over all the multi-indices 0≤l1≤ · · · ≤lt≤m, such that|L|=l1+· · ·+lt ≤mt−t, andR_L(β)∈Z[1/2][β] in (5.9) arepolynomials inβ of degree(mt− |L|)(used with β=κ−k−r).

Note thedifferentiation rule of degreem(see [Sh83], p.466):

∆(fg) = Xm r=0

tr

tρr( ˜∂/∂z)f ·ρ^⋆_m−r( ˜∂/∂z)g .

29

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Example (Siegel-Eisenstein series of odd degree and higher level)

G^∗(z,s;k,ψ,N) (5.10)

=det(y)^sX

c,d

ψ(detc)det(cz+d)^−k|det(cz+d)|^−2s ·

·˜Γ(k,s)LN(k+2s,ψ)





[m/2]Y

i=1

LN(2k+4s−2i,ψ²)



, where

(c,d)runs over all “non-associated coprime symmetric pairs” withdet(c) coprime to N,κ= (m+1)/2, and for m odd theΓ-factor has the form:

˜Γ(k,s) =i^mk2^−m(k+1)π^−m(s+k)Γm(k+s).

We use this series withψ=χ²ψ1ψ2ψ3,k=k2+k3−k1≥2,m=3, κ=m+1

2 =2,[m/2] =1.

30

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Theorem 5.3 [Siegel, Shimura [Sh83], P. Feit [Fei86]]

Theorem

Let m be anodd integersuch that2k >m, and N>1 be an integer, then:

For an integer s such thats=−r ,0≤r ≤k−κ, there is the following Fourier expansion

G^⋆(z,−r) =G^⋆(z,−r;k,ψ,N) = X

Am∋T≥0

a(T,R)q^T, (5.11)

where for s>(m+2−2k)/4in (5.11) the only non-zero terms occur for positive definiteT>0,

Triple products of Coleman’s families Siegel-Eisenstein series

Algebraic differential operators of Maass and Shimura

Fourier coefficients of Siegel-Eisenstein series (continued)

a(T,R) =M(T,ψ,k−2r)·det(T)^k−2r−κQ(R,T;k−2r,r), (5.12) M(T,k−2r,ψ) = Y

ℓ|det(2T)

M_ℓ(T,ψ(ℓ)ℓ^−k+2r) (5.13) polynomials Q(R,T;k−2r,r)are given by (5.9), and for allT>0, T∈Am, is a finite Euler product, in which M_ℓ(T,x)∈Z[x].

Triple products of Coleman’s families Statement of the Main Result

Main Theorem (on p -adic analytic function in four variables)

1)The functionL_f: (s,k1,k2,k3)7→

D

f⁰,E(−r, χ)E D

f⁰,f₀E depends p-adic analytically on four variables

(χ·y_p^r,k1,k2,k3)∈X×B₁×B₂×B₃;

2)Comparison of complex and p-adic values: for all(k1,k2,k3)in an affinoid neighborhoodB=B₁×B₂×B₃⊂X³,satisfying k1≤k2+k3−2: the values at s=k2+k3−2−r coincide with the normalized critical special values

L^∗(f1,k1⊗f2,k2⊗f3,k3,k2+k3−2−r, χ) (6.14) (r=0,· · ·,k2+k3−k1−2),

for Dirichlet charactersχmodNp^v,v ≥1,such that all three corresponding Dirichlet charactersχj haveNp-complete conductors:

33

Triple products of Coleman’s families Statement of the Main Result

Main Theorem (continued)

χ1modNp^v =χ, χ2modNp^v =ψ2ψ¯3χ, (6.15) χ3modNp^v =ψ1ψ¯3χ,ψ=χ²ψ1ψ2ψ₃.

The normalisation of L^∗in (6.14) is the same as in Theorem C below.

3)Dependence on x∈X : let H= [2ordp(λ)] +1. For any fixed (k1,k2,k3)∈Band x =χ·y_p^r the function

x7−→

D

f⁰,E(−r, χ)E D

f⁰,f₀E

extends to a p-adic analytic function of type o(log^H(·))of the variable x∈X .

34

Triple products of Coleman’s families Statement of the Main Result

Outline of the proof

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

f⁰,E(−r, χ)E

=D

f⁰, π_λ(E(−r, χ))E deduced from the definition of the projectorπ_λ: Kerπ_λ:=T

n≥1Im(U_T−λI)ⁿ,Imπ_λ:=S

n≥1Ker(U_T−λI)ⁿ. Notice that the coefficients ofE(−r, χ)∈M(A) dependp-adic analytically on(k1,k2,k3)∈B=B₁×B₂×B₃, where

A=A(B₁×B₂×B₃)is thep-adic Banach algebra of rigid-analytic functions onB.

35

Triple products of Coleman’s families Statement of the Main Result

Interpolation to all p -adic weights:

•At each classical weight(k1,k2,k3)the scalar productD

f⁰,E(−r, χ)E is given by the first coordinate ofπλ(E(−r, χ))with respect to an orthogonal basis ofM^λ(A)containingf₀with respect toHida’s algebraic Petersson producthg,hia:=D

g^ρ|

0 Np−1

0

,hE

, see [Hi90].

Let us extend the linear formℓ(h) = h^f0,hi

h^f0,f0i(defined for classical weights), to Coleman’s type submodule of overconvergent familiesh∈M^λ(A)^†⊂M^λ(A) asthe first coordinate ofhwith respect to someA-basis of eigenfunctions of all (triple) Hecke operatorsTqforq∤Np, having the first basis vector

f₀∈M^λ(A)^†.

The linear formℓcan be characterized as a normalized eigenfunction of the adjoint Atkin’s operator, acting on the dualA-module ofM^λ(A)^†: ℓ(f₀) =1.

In order to extendℓtoh=E(−r, χ), we need to choose a certain representative ofE(−r, χ)in theA-submoduleM^λ(A)^†, which is locally free of finite rank.

36

Triple products of Coleman’s families Statement of the Main Result

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

( A )

Choose a (local) basisℓ¹,· · ·, ℓⁿgiven by sometriple Fourier coefficientsof the dual (locally free of finite rank)A-module M^λ(A)^†∗.

Then define

ℓ=β1ℓ¹+· · ·+βnℓⁿ,

whereβ_i =ℓ(ℓ_i)∈A, andℓ_i denotes the dual basis ofM^λ(A)^†: ℓ^j(ℓ_i) =δ_ij. At eachp-adic weight (k1,k2,k3)∈Blet us define ℓ(E(−r, χ)) :=β1ℓ¹(E(−r, χ))+· · ·+βnℓⁿ(E(−r, χ))(belongs toA), whereβ_i =ℓ(ℓ_i)∈A, andℓⁱ(E(−r, χ))∈Aare certain Fourier coefficients of the seiesE(−r, χ).

37

Triple products of Coleman’s families Statement of the Main Result

Conclusion

There exists an element

E˜(−r, χ)∈M^λ(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(ℓ_n)ℓⁿ(E(−r, χ))

=β1ℓ¹(E(−r, χ)) +· · ·+β_nℓⁿ(E(−r, χ))

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

Thus, the dependence ofℓ(E(−r, χ))∈Aon(k1,k2,k3)∈X³is p-adic analytic.

In order to prove the remaining statements 2), 3), the dependence onx=χ·y_p^r is studied in the next section.

38

Triple products of Coleman’s families Distributions and admissible measures

Distributions and measures with values in Banach modules

A V C(Y,A)

∪

C^loc−const(Y,A)

(a p-adic Banach algebra) (anA-module)

(theA-Banach algebra ofcontinuous functionsonY ) (theA-algebra

oflocally constant functionsonY )

Triple products of Coleman’s families Distributions and admissible measures

Definition (Distributions and measures)

a) AdistributionDon Y with values in V is anA-linear form D:C^loc−const(Y,A)→V, ϕ7→D(ϕ) =

Z

Y

ϕdD.

b) Ameasureµon Y with values in V is a continuousA-linear form µ:C(Y,A)→V, ϕ7→

Z

Y

ϕdµ.

The integral Z

Y

ϕdµcan be defined for any continuous functionϕ, and any bounded distributionµ, using the Riemann sums.

Triple products of Coleman’s families Distributions and admissible measures

Admissible measures of Amice-Vélu

Admissible measures

Lethbe a positive integer. A more delicate notion of an h-admissible measure was introduced in [Am-V] by Y. Amice, J.

Vélu (see also [MTT], [V]):

Definition

a) For h∈N,h≥1letP^h(Y,A)denote theA-module oflocally polynomial functionsof degree<h of the variable

yp:Y →Z^×_p ֒→A^×; in particular,

P¹(Y,A) =C^loc−const(Y,A)

(theA-submodule oflocally constant functions). Let also denote C^loc−an(Y,A)theA-module oflocally analytic functions, so that

P¹(Y,A)⊂P^h(Y,A)⊂C^loc−an(Y,A)⊂C(Y,A).

41

Triple products of Coleman’s families Distributions and admissible measures

Admissible measures of Amice-Vélu

Definition of admissible measures (continued)

b) Let V be a normedA-module with the norm| · |p,V. For a given positive integer h an h-admissible measure on Y with values in V is anA-module homomorphism

Φ :˜ P^h(Y,A)→V

such that for fixed a∈Y and for v→ ∞the followinggrowth conditionis satisfied:

Z

a+(Np^v)

(yp−ap)^h′dΦ˜

_p,V =o(p^{−v(h′−h)}) (7.16) for all h^′=0,1, . . . ,h−1,ap:=yp(a)

The condition (7.16) allows to integratethe locally-analytic functions onY alongΦ˜ using Taylor’s expansions! This means:

there exists a unique extension ofΦ˜ to C^loc−an(Y,A)→V.

42

Triple products of Coleman’s families Distributions and admissible measures

Up–Operator and the method of canonical projection

Using the canonical projection π

We construct ourH-admissible measureΦe^λ:P^H(Y,A)→M(A) out of a sequence of distributionsΦ_r :P¹(Y,A)→M(A)defined on local monomialsy_p^r of each degreer by the rule

Z

Y

χy_p^rdΦe^λ=π_λ(˜E(−r, χ)), whereE˜(−r, χ)∈M=M(A).

HereE˜(−r, χ)takes values in anA-module

M=M(A)⊂A[[q1,q2,q3]][R1,R2,R3]

of nearly holomorphic (overconvergent) triple modular forms overA (for 0≤r≤H−1,H = [2ordpλp] +1), and the formal series

˜E(−r, χ)was constructed in the proof of 1) of Main Theorem.

43

Triple products of Coleman’s families Distributions and admissible measures

Eigenfunctions ofUpand ofU^∗ p.

Functions f

and f

Recall that for any primitive cusp eigenformfj =P_∞

n=1an(f)qⁿ, there is an eigenfunctionf_j,0=P_∞

n=1an(f_j,0)qⁿ∈Q[[q]]ofU=Up

with the eigenvalue α=α⁽_p,j¹⁾ ∈Q(U(f0) =αf0) given by f_j,0=fj−α⁽_p,j²⁾fj|Vp=fj−α⁽_p,j²⁾p^−k/2fj|

p 0 0 1

(7.17) X∞

n=1

an(f_j,0)n^−s = X∞ n=1 p∤n

an(fj)n^−s(1−α⁽_p,j¹⁾p^−s)⁻¹.

Moreover, there is an eigenfunctionf_j⁰ ofU_p^∗ given by f_j⁰=f_j,^ρ₀

k

0 Np

−1 0

, wheref_j,^ρ₀= X∞ n=1

a(n,f0)qⁿ. (7.18) Therefore,UT(f1,0⊗f2,0⊗f3,0) =λ(f1,0⊗f2,0⊗f3,0).

44

Triple products of Coleman’s families

Critical values of theLfunctionL(f1⊗f2⊗f3,s, χ)(Theorem A)

Theorem A (algebraic properties of the triple product)

Assume that k2+k3−k1≥2, then for all pairs (χ,r)such that the corresonding Dirichlet charactersχj have Np-complete conductors, and0≤r≤k2+k3−k1−2, we have that

Λ(f₁^ρ⊗f₂^ρ⊗f₃^ρ,k2+k3−2−r, ψ1ψ2χ) hf₁^ρ⊗f₂^ρ⊗f₃^ρ,f₁^ρ⊗f₂^ρ⊗f₃^ρiT ∈Q where

hf₁^ρ⊗f₂^ρ⊗f₃^ρ,f₁^ρ⊗f₂^ρ⊗f₃^ρiT :=hf₁^ρ,f₁^ρiNhf₂^ρ,f₂^ρiNhf₃^ρ,f₃^ρiN

=hf1,f1iNhf2,f2iNhf3,f3iN.

45

Triple products of Coleman’s families Theorems B-D

Recall: the p -adic weight space and the Mellin transform

Thep-adic weight spaceis the groupX=Homcont(Y,C^×_p)of (continuouos)p-adic characters of the commutative profinite group Y =lim

←−v

(Z/Np^vZ)^∗

The groupX is isomorphic to a finite union of discs U={z∈Cp | |z|^p<1}.

Ap-adicL-functionL_(p):X→Cp is a certain meromorphic function onX. Such a function usually come from ap-adic measure µonY (boundedoradmissiblein the sense of Amice-Vélu, see [Am-V]). Thep-adic Mellin transformofµis given for allx∈X by

L_(p)(x) = Z

Y_N,p

x(y)dµ(y),L_(p):X →Cp

46

Triple products of Coleman’s families Theorems B-D

Theorem B (on admissible measures attached to the triple product:fixed balanced weights case)

Under the assumptions as above there exist aCp-valued measure

˜ µ^λ_f

1⊗f2⊗f3 on Y_N,p, and aCp-analytic function

D_(p)(x,f1⊗f2⊗f3) :Xp→Cp,given for all x∈X_N,pby the integralD_(p)(x,f1⊗f2⊗f3) =R

YN,px(y)d˜µ^λ_f

1⊗f2⊗f3(y),and having the following properties:

Triple products of Coleman’s families Theorems B-D

Theorem B (continued)

(i)for all pairs(r, χ)such thatχ∈X_N,p^tors, and all three

corresonding Dirichlet charactersχ_j have Np-complete conductor (j=1,2,3), and r ∈Zis an integer with0≤r≤k2+k3−k1−2, the following equality holds:

D_(p)(χx_p^r,f1⊗f2⊗f3) =ip

(ψ1ψ2)(2)Cχ^{4(k2+k3−2−r)}

G(χ1)G(χ2)G(χ3)G(ψ1ψ2χ1)λ^2v_p Λ(f₁^ρ⊗f₂^ρ⊗f₃^ρ,k2+k3−2−r, ψ1ψ2χ)

hf₁⁰⊗f₂⁰⊗f₃⁰,f1,0⊗f2,0⊗f3,0iT,Np

where v =ordp(C_χ), G(χ)denotes the Gauß sum of a primitive Dirichlet characterχ0attached toχ(modulo the conductor ofχ0),

Triple products of Coleman’s families Theorems B-D

Theorem B (continued)

(ii)ifordpλp=0then the holomorphic function in (i) is abounded Cp-analytic function;

(iii)in the general case (but assuming thatλp6=0) the holomorphic function in (i) belongs to thetypeo(log(x_p^H))with H= [2ordpλp] +1and it can be represented as the Mellin transform of the H-admissibleCp-valued measureµ˜^λ_f

1⊗f2⊗f3 (in the sense of Amice-Vélu) on Y

(iv)Let k=k2+k3−k1≥2. If H≤k−2then the functionD_(p) is uniquely determined by the above conditions (i).

49

Triple products of Coleman’s families Theorems B-D

Theorem C (on p -adic measures for families of triple products)

Put H= [2ordp(λ)] +1. There exists a sequence of distributions Φr on Y with values inM=M(A)giving an H-admissible measure Φ˜^λwith values inM^λ⊂M with the following properties:

There exists anA-linear formℓ=ℓf₁⊗f₂⊗f₃,λ:M(A)^λ→A(given by (10.21), such that the composition

˜

µ= ˜µf₁⊗f₂⊗f₃,λ:=ℓf₁⊗f₂⊗f₃,λ( ˜Φ^λ)

is an H-admissible measure with values inA, and for all(k1,k2,k3) in the affinoid neighborhoodB=B₁×B₂×B₃,as above, satisfying k1≤k2+k3−2

50

Triple products of Coleman’s families Theorems B-D

we have that the evaluated integrals ev_(k1,k2,k3)

(ℓf₁⊗f₂⊗f₃,λ)( ˜Φ^λ)(y_p^rχ)

on the arithmetical chracters y_p^rχcoincide with the critical special values

Λ^∗(f1,k1⊗f2,k2⊗f3,k3,k2+k3−2−r, χ)

for r=0,· · ·,k2+k3−k1−2, and for all Dirichlet characters χmodNp^v,v≥1, with all three corresonding Dirichlet characters χj given by (6.15), having Np-complete conductors. Here the normalisation ofΛ^∗includes at the same time certain Gauss sums, Petersson scalar products, powers ofπ and ofλ(k1,k2,k3), and a certain finite Euler product.

51

Triple products of Coleman’s families Theorems B-D

The p -adic Mellin transform and four variable p -adic analytic functions

Anyh-admissible measureµ˜ onY with values in ap-adic Banach algebra Acan be caracterized its Mellin transformL_µ˜(x)

L_µ˜:X→A, defined byL_µ˜(x) =R

Yx(y)dµ(y),˜ where x∈X, L_µ˜(x)∈A,

Key property ofh-admissible measuresµ:˜ its Mellin transformL_µ˜is analyticwith values inA.

LetA=A(B) =A₁⊗ˆA₂⊗ˆA₃=A(B₁) ˆ⊗A(B₂) ˆ⊗A(B₃)denote again the Banach algebraAwhereBis an affinoid neighbourhood around(k1,k2,k3)∈X³(with a given integerk and Dirichlet characterψmodN).

52