On p-adic integration in spaces of modular forms and its

applications

A. A. Panchishkin∗

Institut Fourier, B.P.74, 38402 St.–Martin d’Hères, FRANCEe-mail : panchish@mozart.ujf-grenoble.fr, FAX: 33 (0) 4 76 51 44 78

Abstract

The purpose of this course is to give an introduction to the theory of p-adic integration with

values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show

that very general p-adic families of modular forms can be constructed as moments of certain p-adic

measures on a profinite group Y = lim←−

Yi with values in a formal q-expansion ring like Zp[[qB ]] where

B is an additive semi-group, and qB = {qξ |ξ ∈ B} the corresponding formally written multiplicative

semi-group (for example B = Bn = {ξ = tξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group,

important for the theory of Siegel modular forms).

We discuss some applications of this theory to the construction of certain new p-adic fami-

lies of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical

polynomials. . .). Main sources of this theory are:

• Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et

fonctions zêta p-adiques, LNM 350 (1973) 191-268) [Se73].

• Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory

of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]).

• Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE].

As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the Rankin-

Selberg method and the p-adic integration in a Banach algebra A.

∗An introductory cours given on November 29 in POSTECH (Pohang, Korea)

0 Introduction

Let p be a prime number (we often assume p≥ 5).

There are two different ways of introducing p-adic modular forms: the first approach uses formal

q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation

of Galois representations attached to classical automorphic forms. The first approach was extensively

developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct

p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion

method a typical p-adic family ϕ of modular (automorphic) forms is an element of the Serre ring: ϕ ∈Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois

representations of type ρ : Gal(Q/Q)→ GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]).

These two theories are essentially equivalent if we start from holomorphic automorphic forms on the

group G = GL2 over a totally real field, but in other cases there is no direct link between ϕ and ρ. On

the other hand there exist interesting examples of p-adic L-functions Lϕ,p and Lρ,p attached to ϕ and

to ρ. In general Lϕ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions.

If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see

[Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type

conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations.

As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg

method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated

in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows:

Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ =

ordp(αp(k′)) > 0, to construct a two variable p-adic L-function interpolating on k′ the Amice-Vélu

p-adic L-functions Lp(fk′ ).

Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures

come from Eisenstein distributions with values in certain Banach A-modules M† = M†(N ; A) of families

of overconvergent forms over A.

2

Contents

0 Introduction 2

1 Abstract Kummer congruences and Λ-adic Eisenstein series 3

2 Algebraic Petersson product of Λ-adic modular forms 12

3 Λ-adic Siegel-Eisenstein series 17

4 Λ-adic Klingen-Eisenstein series 22

5 Λ-adic theta functions with spherical polynomials 25

6 p-adic families of automorphic representations 29

7 Application: a solution of the problem of Coleman-Mazur 31

8 General L-functions of two variables and admissible measures 35

1 Abstract Kummer congruences and Λ-adic Eisenstein series

A preliminary definition of a p-adic family of modular forms is the following: {fk} is an infinite set

of modular forms fk =∑

n ≥ 0ak(n)qn ∈ Q[[q]] (parametrized by the weight k), whose Fourier coefficients

depend p-adic analytically on the weight k. If we fix embeddings i∞ : Q →֒ C, ip : Q →֒ Cp (Cp = Q̂p

is the Tate field) then this amounts to analyticity of all maps k 7→ ak(n) (where k is viewed as a p-adic

parameter).

There are various approaches to construction of such families:

1) The technique of Hida and Wiles of Λ-adic modular forms where Λ = Zp[[T ]] is the Iwasawa algebra:

each element A(T ) =∑

n ≥ 0anT

n ∈ Λ converges on the unit disc U = {t ∈ Cp

|t|p < 1} to

a bounded Cp-analytic function with a finite number of poles (in view of the p-adic Weierstrass

preparation theorem, see [Ko]). We have therefore for each k ≥ 1 a specialization map:

A(T ) 7→ Ak = A((1 + p)k − 1) ∈ Zp,(a homomorphism κk : Λ→ Zp

given by κk(1 + T ) = (1 + p)k).

In order to introduce Λ-adic modular form consider the Serre ring Λ[[q]] (i.e. the ring of all formal

q-expansions with coefficients in Λ = Zp[[q]], and for each element F =∑

n ≥ 0A(n, T )qn ∈ Λ[[q]] and

for each k ≥ 1 we have a ring homomorphism

κk : Λ[[q]]→ Zp[[q]], defined by

κk(F ) =∑

n ≥ 0

Ak(n)qn, where Ak(n) = A(n, (1 + p)k − 1) ∈ Zp .

3

Let χ be a Dirichlet character modulo N and let us fix a finite extension O of Zp containing the

values of χ, and put ΛO = O[[T ]]. Consider the Serre ring ΛO[[q]] of all formal q-expansions with

coefficients in ΛO.

Definition 1.1. An element F =∑

n ≥ 0A(n, T )qn ∈ ΛO[[q]] is called a ΛO-adic modular form in the

sense of Wiles [Wi88] of level N and character χ if for all k ≫ 0 the element

Fk =∑

n ≥ 0

A(n, (1 + p)k − 1)qn ∈ O[[q]]

represents the Fourier expansion of a classical modular form: F ∈Mk(Γ0(Np), χω−k), where ω : (Z/pZ)∗ →

µp−1 ⊂ Q∗ →֒ C∗ is the Teichmüller character.

Let us denote by M(N,χ,ΛO) the ΛO-submodule of ΛO[[q]] generated by all such elements.

2) the second approach to p-adic families of modular forms uses the notion of the p-adic integration

with values in spaces of modular forms.

Consider any profinite group Y = lim←−Yi (for example, Y = Zp, Y = Γ = 1 + pZp = 〈1 + p〉,Y = Gal(Q/Q) etc.).

The completed group ring O[[Y ]] = lim←−O[Yi] is a compact ring which generalizes the Iwasawa algebra

ΛO, and it may be viewed as the O-algebra of all O-valued distributions on Y ; for the natural

projection homomorphisms πi : Y → Yi put Uij = π−1i (yj) (yj ∈ Yi) then Uij generate the profinite

topology of Y and a distribution µ on Y corresponds to an element f = {fi} ∈ O[[Y ]] where

fi(yj) = µ(Uij).

Recall that a distribution µ on Y with values in O is a finite-additive map on open-compact subsets

U ⊂ Y :

µ :

{open-compact sets

U ⊂ Y

}−→ O

The natural multiplication in O[[Y ]] corresponds to the convolution of distributions:

(µ1 ∗ µ2)(π−1i (y)) =

∑

y=z·w∈Yi

µ1(π−1i (z)) · µ2(π

−1i (w))

4

Examples.

a) ΛO = O[[T ]] ∼= O[[Γ]], 1 + T 7→ 1 + p ∈ Γ.

Any distribution µ ∈ Distr(Zp,Zp) is determined by the power series

Aµ(T ) =

∫

Zp

(1 + T )xdµ(x) ∈ Zp[[T ]] =⇒ Aµ((1 + p)k − 1)) =

∫

Zp

(1 + p)kxdµ(x)

b) ΛO = O[[Z∗p]]∼= ΛO ⊗O O[∆] where Z∗

p = Γ×∆ (a direct product, where ∆ = Z∗torsp ).

For a continuous group homomorphism κ : Y → O∗ we obtain an O-algebra homomorphism

κ : O[[Y ]] → O which provides a group-theoretic interpretation of the above specialization maps:

κk(y) = yk for all x ∈ Z∗p.

In order to define more general p-adic families of modular forms, consider a family P = {P} ⊂ C(Y,O)

of continuous functions on Y with values in O.

Definition 1.2. A p-adic family of modular forms parametrized by P is an element F of the generalized

Serre ring

O[[Y ]][[q]] = Distr(Y,O[[q]])

such that if F =∑

n ≥ 0µnq

n, µn ∈ Distr(Y,O) = O[[Y ]], then for almost all P ∈ P the integral

FP =

∫

Y

PdF =∑

n ≥ 0

(∫

Y

Pdµn

)qn ∈ O[[q]]

represents the Fourier expansion of a classical modular form of weight k = k(P ), level N = N(P ) and

character χ = χP (this is an integral with values in modular forms over O; we integrate a continuous

function P with respect to a measure dF taking values in the O-submodule of O[[Y ]][[q]] generated by the

Fourier expansions of classical modular forms).

Note that the parameter P could include both classical parameters: the weight k and the character

ε mod pα.

Let us use the notation MP(O[[Y ]]) for the O[[Y ]]-submodule of O[[Y ]][[q]] generated by all such F .

One verifies that in the case of Y = Γ, ΛO = O[[Γ]], and

P ={P = Pk,ε : 7−→ ε(u)uk, ε mod pα, u ∈ Γ

},

5

this leads to the first definition (we assume that O contains the values of all Dirichlet characters ε).

The first example of p-adic families is related to the classical Eisenstein series ((see in Serre, [Se73]) :

consider

G∗k =

ζ∗(1− k)2

+

∞∑

n=1

σ∗k−1(n)qn, σ∗

k−1(n) =∑

d|n(d,p)=1

dk−1, where

ζ∗(s) = ζ(s)(1 − p−s) =∑

n=1(p,n)=1

n−s denotes the Riemann zeta function

with p-Euler factor removed.

Then we prove in Theorem 1.2. below that fk = (1− ck)G∗k is a p-adic family on Z∗

p parametrized by

the set P = {y 7→ yk, k ≥ 4} of p-adic characters of Z∗p: there exists a measure

f =1

2µ(c) +

∑

n ≥ 0

µn,cqn ∈ Λ[[q]],Λ = Distr(Z∗

p,Zp)

on Z∗p with values in Zp[[q]] such that

∫

Z∗p

xkdf =1

2

∫

Z∗p

xkdµ(c) +∑

n ≥ 0

(∫

Z∗p

xkdµn,c

)qn = (1− ck)G∗

k .

An advantage of the second approach is a possibility to use the technique of abstract Kummer con-

gruences, following [Ka78]. Consider a complete p-adic ring A = lim←−N

A/pNA and consider any family

{ϕi}i∈I ⊂ C(Y,Zp) which generate a dense Qp-linear subspace.

Theorem 1.1. Let {ai} ⊂ A a collection of elements of A parametrized by the same set of indices I

that we used for the family of continuous functions {ϕi}. Then the existence of a unique distribution

µ ∈ Distr(Y,A) with given integrals ai =∫

Y

ϕidµ is equivalent to the validity of the following congruences:

for all N ∈ N and for each finite linear combination∑i

biϕi, bi ∈ Zp

(∀ y ∈ Y

∑

i

biϕi(y) ≡ 0 mod pNA =⇒∑

i

biai ≡ 0 mod pN

)(1.1)

Proof of Theorem 1.1: (see [Ka78])

1) If µ exists then∫

Y

∑biϕidµ =

∑biai ≡ 0(pNA) because

∑biϕi is divisible by pN in the ring C(Y,A).

2) In order to construct µ out of ai ∈ A, we use the equivalent definition of an A-distribution as an

A-measure (a Zp-linear form on C(Y,Zp) with values in A): we need to construct such a linear form

µ : C(Y,Zp)→ A with the condition ∀i ∈ I∫

Y

ϕidµ = ai

6

For any ϕ ∈ C(Y,Zp) let us choose an approximation∑biϕi. Then

ϕ = lim∑

i

biϕi ⇒∫ϕdµ = lim

∑

i

biai ∈ A.

We see that the limit is well defined in view of the congruences (1.1).

Example. Mahler’s series ([Hi93], [PaLNM])Let Y = Zp, A = Zp,

C(Zp,Zp) =

{ϕ(x) =

∑

n ≥ 0

an

(x

n

), |an|p → 0

}

Note that

(x

n

)=x(x− 1) . . . (x− n+ 1)

n!: Zp → Zp,

ϕ(m) =

m∑

n=0

an

(m

n

), an =

n∑

i=0

(−1)n−iϕ(i)

(n

i

)

We see that every µ is determined by any {bn} ⊂ Zp, bn =

∫

Zp

(x

n

)dµ(x) by the formula

∫

Zp

ϕ(x)dµ =

∑n ≥ 0

anbn, and ∀m ∈ N,

m∑

n=0

an

(m

n

)≡ 0 mod pN ⇐⇒ ∀n ∈ N, an ≡ 0 mod pN

Any such measure µ is characterized by its Amice transform :

Aµ(T ) =

∫

Zp

(1 + T )xdµ =∑

j ≥ 0

∫

Zp

(x

j

)dµT j =

∑

j ≥ 0

bjTj ∈ Zp[[T ]],

where

(x

j

)=x(x − 1) · · · (x− j + 1)

j!. This gives the Iwasawa isomorphism:

{Zp-valued measures on Zp} ∼→ Zp[[T ]], Aµ1∗µ1(T ) = Aµ1 (T )Aµ2(T )

and we have a very explicit formula:

µ(a+ (pN )) =1

pN

∑

ζ,ζpN =1

ζ−a

∫

Zp

ζxdµ =1

pN

∑

ζ,ζpN =1

ζ−aAµ(ζ − 1)dµ :

by the orthogonality relations δa+(pN )(x) =1

pN

∑

ζ,ζpN =1

ζ−aζx and by definition of Aµ

Aµ(ζ − 1) =∫

Zp(1 + (ζ − 1))xdµ =⇒∑

ζ,ζpN =1 ζ−aAµ(ζ − 1)dµ ≡ 0 mod pN .

7

Let us describe in more detail the example classical Eisenstein series: consider Γ = SL2(Z) ∋ γ =(a bc d

),

P =

{γ =

(a b

0 d

)∈ Γ

}, k ≥ 4, Im(z) > 0 ,

Ek(z) =∑

γ∈P\Γ

1|kγ =∑

(c,d)=1,c>0c=0,d=1

(cz + d)−k = 1 +2

ζ(1 − k)

∞∑

n=1

σk−1(n)qn ,

q = exp(2πiz), σk−1(n) =∑

d|n

dk−1, ζ(s) =

∞∑

n=1

n−s =∏

ℓ

(1− ℓ−s)−1 (Re(s) > 1)

ζ(1 − k) = −Bk

k,

∞∑

k=0

Bktk

k!=

t

et − 1= “eBt”

(E4 = 1 + 240

∞∑

n=1

σ3(n)qn, E6 = 1− 504

∞∑

n=1

σ5(n)qn, . . .

).

Let us normalize these series by: Gk = ζ(1−k)2 Ek = −Bk

2k +∞∑

n=1σk−1(n)qn and put

G∗k(z) = Gk(z)− pk−1Gk(pz) then

G∗k =

ζ∗(1− k)2

+

∞∑

n=1

σ∗k−1(n)qn, σ∗

k−1(n) =∑

d|n(d,p)=1

dk−1, where

ζ∗(s) = ζ(s)(1 − p−s) =∑

n=1(p,n)=1

n−s denotes the Riemann zeta function

with p-Euler factor removed.

Theorem 1.2.

a) Let k ≡ k′ mod (p− 1)pN−1 then G∗k ≡ G∗

k′ mod pN in Q[[q]] for all k 6≡ 0 mod (p− 1).

b) Let k ≡ k′ mod (p− 1)pN−1 then for any c ∈ Z, (c, p) = 1, c > 1 we have that (1 − ck)G∗k ≡

(1 − ck′

)G∗k′ mod pN (without restriction on k).

c) fk = (1 − ck)G∗k is a p-adic family on Z∗

p parametrized by the set P = {y 7→ yk, k ≥ 4} of p-adic

characters of Z∗p.

Proof of Theorem 1.2: The statements a) and b) follow from c). In order to show c) we put fk =∑

n ≥ 0ak(n)qn

The case n > 0: ak(n) = (1− ck)∑d|n

(d,p)=1

dk−1 =∫

Zp

xkdµn,c(x) where

8

µn,c = (δ1 − δc) ∗∑d|n

(d,p)=1

d−1 · δd, δc(U) =

{1, if c ∈ U,0, if c /∈ U,

, the Dirac distribution at c.

Note that for any µ(1), µ(2) ∈ Distr(Z∗p,O), and for any character P ∈ Homcont(Z

∗p,O

∗) we have that∫

Z∗p

Pd(µ(1) ∗ µ(2)) =∫

Z∗p

Pdµ(1) ·∫

Z∗p

Pdµ(2).

The case n = 0: ak(0) = (1− ck)ζ∗(1− k) will be treated using the classical Kummer congruences given

by the following

Theorem 1.3 (Kummer). Put ζ(c)(p)(−k) = (1 − ck+1)(1 − pk)ζ(−k), k ≥ 0, and let h(x) =

∑i

αixi ∈ Z[x]

such that h(a) ≡ 0 mod pN for all α ∈ Z∗p. Then

∑i

αiζ(c)(p)(−i) ≡ 0 mod pN .

Proof of Theorem 1.3 [Ka78]: uses the sums of powers Sk(M) =M−1∑n=1

nk, the Bernoulli numbers Bk, and

and the Bernoulli polynomial Bk(x):

Sk(M) =

M−1∑

n=1

nk =1

k + 1[Bk+1(M)−Bk+1], where

∞∑

m=1

Bk

k!tk =

tet

et − 1and Bk(x) =

k∑

i=0

(k

i

)Bix

k−i.

Consider the regularized sums of powers

S∗k(pN ) =

pN−1∑

n=1p∤n

nk = Sk(pN )− pkSk(pN−1) ,

and express the Bernoulli numbers in terms of Sk(N):

Bk+1 = limN→∞

1

pNSk+1(p

N ),

(the p–adic limit) which follows directly from the above formula for Sk(pN ).

For each n with (p, n) = 1 we have the congruence h(n) ≡ 0(modpN ), and

limN→∞

1

pNS∗

k+1(pN ) = lim

N→∞

1

pN[Sk+1(p

N )− pk+1Sk(pN−1)] =

limN→∞

1

pNSk+1(p

N )− pk limN→∞

1

pNSk+1(p

N−1) = (1− pk)Bk+1.

Let us substitute ζ(−k) = −Bk+1

k + 1then

ζ(c)(p)(−k) = (ck+1 − 1)(1− pk)

Bk+1

k + 1≡ Sk+1(p

M )

pM· (c

k+1 − 1)

k + 1mod pN (1.2)

9

(for a sufficiently large M ≥N). The right hand side of (1.2) transforms by definition to

pM−1∑

n=1p∤n

(cn)k+1 − nk+1

pM · (k + 1)=

pM−1∑

n=1p∤n

(cn)k+1 − nk+1c

pM · (k + 1)(1.3)

where n 7→ nc is a permutation of {1, 2, . . . , pM − 1} given by nc ≡ nc (mod pM ). Let us substitute

cn = nc + pM tn, tn ∈ Z into (1.3):

(nc)k+1 − nk+1c

pM · (k + 1)≡ tn · nk

c mod pM

therefore ζ(c)(p)(−k) ≡

∑

n=1p∤n

t·ckc mod pM where tn = t(n, c) is independent of k. In order to finish the proof,

let us substitute this congruence into the linear combination of Theorem 1.3 using h(x) ≡ 0 mod pN :∑i

αiζ(c)(p)(−i) =

∑n=1p∤n

tn · h(nc) ≡ 0 mod pN .

Corollary 1 (p-adic continuity of ζ(c)(p)(−k) in a progression mod(p− 1)). If h(x) = xk − xk′

, k ≡ k′ mod

(p− 1) pN−1 then

ζ(c)(p)(−k) ≡ ζ

(c)(p)(−k

′) mod pN .

Proof of Corollary 1: We have that h(a) ≡ 0 mod pN by Euler’s theorem: aϕ(pN ) ≡ 1 (mod pN),

(a, p) = 1.

Corollary 2 (Mazur). There exists a unique measure µ(c) on Z∗p such that for all k ≥ 1

∫

Z∗p

xkdµ(c) =

ζ(c)(p)(1− k) = (1− ck)(1 − pk−1)ζ(1 − k).

Proof of Corollary 2: is implied by Theorem 1.1: the values∫

Z∗p

xk+1dµ(c) = ζ(c)(p)(−k) satisfy the abstract

Kummer congruences in view of Theorem 1.3.

Let us show that Theorem 1.3 implies Theorem 1.2 c): put

f =1

2µ(c) +

∑

n ≥ 0

µn,cqn ∈ Λ[[q]],Λ = Distr(Z∗

p,Zp) .

Then ∫

Z∗p

xkdf =1

2

∫

Z∗p

xkdµ(c) +∑

n ≥ 0

(∫

Z∗p

xkdµn,c

)qn = (1− ck)G∗

k .

10

Remark. The use of a regularizing integer c > 1, c ∈ Z∗p in Theorem 1.2 can be avoided, if we con-

sider modular forms with denominators: let Y = Z∗p, O ⊃ Zp, LO = QuotΛO, LO = QuotΛO, the

complete quotient ring of the completed group ring ΛO = O[[Z∗p]]. Let us denote by M(N,χ,ΛO) the

ΛO-submodule of ΛO[[q]] generated by all F =∑

n ≥ 0µnq

n ∈ ΛO[[q]] such that for all k ≫ 0∫

Z∗p

xkdF =

∑n ≥ 0

(∫

Z∗p

xkµn

)qn ∈ O[[q]] is in Mk(Np, χ, A) where A is a subring of Q such that ip(A) ⊂ O. Define

M(N,χ,LO) = M(N,χ,ΛO)⊗ΛO

LO then the series f of Theorem 1.2 is an element of M(1, 1,ΛZp)

depending on c, and 1 − ck =∫

Z∗p

xk(δ1 − δc) (δc denotes the Dirac distribution at c ∈ Z∗p), δc(U) =

{1, if c ∈ U0, if c /∈ U.

The element (δ1 − δc)−1 ∈ LZpand G∗ = f ⊗ (δ1 − δc)−1 is an element of M(1, 1,LZp

)

independent on the choice of c. It has the property κk(G∗) = G∗k where κk is the unique extension of

κk : ΛZp→ Zp to LZp

(we apply κk coefficient-by-coefficient to G∗ ∈M(1, 1,LZp)).

11

2 Algebraic Petersson product of Λ-adic modular forms

We show first how to construct p-adic families of cusp forms starting from a given cusp form and the

Λ-adic Eisenstein series of §1. Then we describe Hida’s algebraic scalar product of Λ-adic modular forms.

Recall that the Λ-adic Eisenstein series G∗ ∈M(1, 1,L) of §1 was constructed as (1− δc)−1 ·f , where

f =µ(c)

2+

∞∑

n=1

µn,cqn ∈ Λ[[q]] =⇒ G∗ = (1− δc)−1 · f ∈ L[[q]].

Theorem 2.1 (Kubota-Leopoldt-Iwasawa [KuLe], [Iw]).

a) There exists a unique element g ∈ L such that for all k ≥ 1 κk(g) = ζ∗(1 − k). It has the property

(1 − δc) · g = µ(c) ∈ Λ = Distr(Z∗p,Zp).

b) For each i mod p− 1 there exists a unique p-adic meromorphic function ζp,i(s) on s ∈ Zp r{1} such

that ζp,i(1 − k) = ζ∗(1 − k) for all k ≡ i mod (p− 1) k ≥ 1. The function ζp,i(s) is holomorphic if

i 6≡ 0 mod p− 1 and it vanishes identically if i is odd (i.e., 1− k is an even negative number).

Proof of 2.1: a) is a direct consequence of Theorem 1.3, because of the equality∫

Z∗p

xkµ(c) = ζ(c)(p)(1− k) = (1− ck)ζ∗(1− k), k ≥ 0 .

In order to prove b) we may put

ζp,i(s) =

∫

Z∗p

〈x〉1−s ωi(x)µ(c)(x) × (1− 〈c〉i−s ωi(c))−1, ζp,i(1− k) = ζ∗(1− k) (2.1)

where ω(x) · 〈x〉 = x ∈ Z∗p, and we put s = 1− k, k ≡ i(p− 1).

Remark. If k ≡ i(p− 1) and k is even, k > 1, then ζ(−k) = 0 and ζp,1+i(s) vanishes identically. However

one can modify this definition using the functional equation of ζ(s) (P. Colmez [Colm98]): one takes

instead of ζp,1+i(s) the function ζ+p,i(s) defined by

ζ+p,i(k) =

Γ(k)(2π√−1)k (1− p)ζ(k) ,

for k even, k ≡ i mod p− 1, k > 1.

If one accepts this modification then the resulting function ζ+p,i(s) will satisfy a p-adic functional

equation described in [Colm98]: ζ+p,i(s) = 2ζp,i(1− s).

12

Let us construct p-adic families of cusp forms starting from a given modular form f ∈Mm(Γ0(N), χ, A)

and the Λ-adic Eisenstein series G∗ ∈M(1, 1,L) of §1 given by

κk(G∗) = G∗k =

(1 − pk−1)ζ(1 − k)2

+∑

n ≥ 1

(∑

d|np∤d

dk−1

)qn (2.2)

Here A is a subring of Q containing the values of χ viewed as a subring of O, a finite ring extension of

Zp. Then L ⊂ LO and Mm(Γ0(N), χ, A) is contained in M(N,χ,LO), M(1, 1,L) ⊂ M(N, 1,LO). Note

that f ·G∗ ∈ LO[[q]] is a formal q-expansion such that κk(f ·G∗) = f ·G∗k ∈Mk+m(Γ0(N), χ,KA) where

KA = QuotA.

Let us consider an (O-algebras) homomorphism σ−m : LO → LO obtained as extension of the group

homomorphism σ−m : Z∗p → Z∗

p such that σ−m(y) = y−m, y ∈ Z∗p. Then κk ◦ σ−m = κk−m and

σ−m(f ·G∗) defines an element of M(N,χ,LO) (the convolution product f ∗G∗, see [Hi93, p. 200].

Definition. Let S(N,χ,Λ) be the ΛO-submodule of cusp forms, and put S(N,χ,LO) = S(N,χ,ΛO)⊗LO.

If f ∈ Sm(Γ0(N), χ, A) is a fixed cusp form then obviously

f ∗G∗ ∈ S(N,χ,LO).

Hecke operators on Λ-adic modular forms. Recal first the classical case: if f =∑

n ≥ 0a(n, f)qn

∈Mk(Γ0(N), χ) then for all n ∈ N coprime with N we have that f |T (n) ∈Mk(Γ0(N), χ) where

a(m, f |T (n)) =∑

d|(m,n)

dk−1χ(d)a(mnd2

, f)

(2.3)

Now let f ∈M(N,χ,LO) be a (meromorphic) Λ-adic form, and p ∤ n. Define

a(m, f |T (n)) =∑

d|(m,n)

δd · d−1χ(d)a(mnd2

, f)

where δd ∈ Z∗p ⊂ LO is characterized by κk(δd) = dk and we have that

κk(a(m, f |T (n)) =∑

d|(m,n)

dk−1 · χ(d)a(mnd2

,κk(f))

=⇒ f |T (n) ∈M(N,χ,LO). (2.4)

13

Algebraic Petersson product: an abstract setting.

Let S ⊂ A[[q]] be an A-module of modular forms over a Noetherian integral ring A. Assume that S is

of finite type, and let h(S) be the commutative Hecke algebra of S over A (the A-subalgebra of EndAS

generated by T (n)). Assume the following properties.

(1) S ∼= HomA(h(S), A) via 〈f, h〉 = a(1, f |h), f =∞∑

n=1a(n, f)qn ∈ S, h ∈ h(S).

(2) h(S)⊗A K is semisimple over K = QuotA.

Semisimplicity implies that there exists a non-degenerate trace-pairing on D = h(S) ⊗A K given by

(h, g) = TrD/K(hg). It follows that there exists a natural isomorphism i : D ∼−→D∗, (where D∗ is the

dual K-vector space of K) hence we obtain the dual pairing

( · , · )A : S(K)× S(K)→ K given by (f, g)A = i−1(f)(g) .

Since we have for all h ∈ h(S)

(f |h, g)A = (f, g|h)A (h is self-adjoint) (2.5)

(the pairing is called the algebraic Petersson product).

Geometric meaning: let f |h = λ(h)f , λ : h(S)→ A be an A-algebra homomorphism. If a(1, f) = 1 then

c(f, g) =(f, g)A

(f, f)A∈ K is well defined and coincides with the coefficient of f if we express g as a linear

combination of normalized eigenforms:

if g =∑

i

c(fi, g) · fi, f1 = f then c(f, g) = c(f1, g)

(By (2), S(K) has a basis of normalized eigenforms under a finite extension of K if necessary).

Examples

1) The classical case: put Mk(Γ0(N), χ;A) = Mk(Γ0(N), χ; Z[χ]) ⊗Z[χ]

A for any subring A of C con-

taining the values of χ : (Z/NZ)∗ → C∗, and define the following A-modules: Sk(Γ0(N), χ;A) ⊂Mk(Γ0(N), χ;A), Hk = h(Mk(Γ0(N), χ;A),

hk = h(Sk(Γ0(N), χ;A)) .

14

Theorem 2.2 (of duality, [Hi93, p. 142]). For any A ⊃ Z[χ],

HomA(hk(Γ0(N), χ;A)) ∼= Sk(Γ0(N), χ;A)

Theorem 2.3 (of semi-simplicity, [Hi93, p. 145]). If χ is primitive modulo pr (or r = 0) then

Hk(Γ0(pr), χ; C) is a semisimple C-algebra and Mk(Γ0(p

r), χ; C) is spanned by common eigenforms

f of all Hecke operators such that f |T (n) = a(n, f) · f .

2) The Λ-adic case can be treated using Hida’s ordinary forms. Let O ⊃ Zp[χ], ΛO = O[[T ]].

One can define the ordinary projection operator e ∈ H(N,χ,ΛO) by

Lemma 2.4 ([Hi93, p. 201]). For any commutative O-algebra A of finite rank and for any x ∈ Athe limit e(x) = lim

n→∞xn! exists in A and e2(x) = e(x).

Define ek = limn→∞

U(p)n! ∈ Hk(Γ0(N), χ; O) assuming k fixed, p divides N ,

∑

n ≥ 0

a(n, f)qn|U(p) =∑

n ≥ 0

a(pn, f)qw .

Then one verifies that ek can be extended to an element e ∈ H(N,χ,ΛO) in such a way that

κk(ef) = ekκk(f) for all k. Let us define: Hord(N,χ,Λ) = e(H(N,χ,Λ)), hord(N,χ,Λ) =

e(h(N,χ,Λ)), Mord(N,χ,Λ) = eM(N,χ,Λ), Sord(N,χ,Λ) = eS(N,χ,Λ), Λ = ΛO.

Theorem 2.5 (Wiles, see [Hi93, p. 209]). The ordinary Hecke algebra over Λ hord(N,χ,Λ) is a free

Λ-module of finite rank.

Theorem 2.6 (Λ-adic duality, [Hi93, p. 218]). For N = pα

Sord(N,χ,Λ) ∼= HomΛ(hord(N,χ,Λ),Λ)

Theorem 2.7 (Λ-adic semisimplicity). Hord(N,χ,Λ) (resp. hord(N,χ,Λ)) is reduced, i.e. Hord(N,χ,L)

(resp. hord(N,χ,L)) is a semisimple L-algebra of finite rank.

Corollary 2.8. We have that Hord(N,χ,L) ∼=∏K

K is a product of finite extensions K/L.

15

Theorem 2.9. For a suitable finite extension K ⊂ L of L, the K-vector spaces Mord(N,χ,K) and

Sord(N,χ,K) have a (finite) basis fj =∑

a ≥ 0

a(n, fj)qn consisting of common eigenfunctions of

Hord(N,χ,K) (resp. hord(N,χ,K)).

(a) If one normalizes such a basis fj so that a(1, fj) = 1, then fj ∈ Sordd(N,χ, I), I the integral

closure of ΛO in K.

(b) If g =∑c(fj , g) · fj then c(fj , g) =

(fj ,g)(fj ,fj) ∈ K.

Theorem 2.10 (fundamental theorem of the theory of ordinary forms).

a) rkOHordk

(Γ0(p

α), χω−k; O)

= rkOMordk

(Γ0(p

α), χω−kO)

= rkOMord2 (Γ0(p

α), χω−2; O) = rkΛMord(pα, χ,ΛO), k ≥ 2 .

b) The same assertion holds for hordk and Sord

k .

In other words, the rank rkOHordk

(Γ0(p

α), χω−k)

is independent of k and each classical ordinary cusp

eigenform can be included into a p-adic family, see [Hi93, Chapter VII].

Note that ΛO = Distr(Γ,O) and for each f ∈M(pα, χ,ΛO)

fk = fT=(1+p)k−1

= κk(f) where κk : O[[Γ]]→ O, given by

κk(〈x〉) = 〈x〉k for all 〈x〉 ∈ Γ, fk =

∫

Γ

〈x〉k df .

If we consider ΛO = ΛO ⊗O O[∆] then for each f ∈M(pα, χ,ΛO)

fk =

∫

Γ

〈x〉k df =1

|∆|

∫

Γ×∆

〈x〉k · ω(x)k · ω−kdf =1

|∆|

∫

Z∗p

xk · ω−kdf ∈M(pα, χω−k; O)

and {fk} defines a family from Y = Z∗p with P = {x 7→ xk · ω−k}, such that 1

|∆|(κkω−k)(f) = fk where

κk(x) = xk = 〈x〉k ω(x)k is a p-adic character on Z∗p which extends the character κk(〈x〉) = 〈x〉k of the

subgroup Γ ⊂ Z∗p.

16

3 Λ-adic Siegel-Eisenstein series

Involuted Eisenstein series. Recall first the elliptic modular case: for a Dirichlet character χ mod N , and

k > 2, χ(−1)k, the following Eisenstein series is defined

Gk(z, χ) =L(1− k, χ)

2+

∞∑

n=1

σk−1,χ(n)qn ∈Mk(Γ0(N), χ,Q(χ)) .

If χ is primitive, then

Gk(z, χ) =L(1− k, χ)

2

∑

γ∈PrΓ0(N)

χ(d)(cz + d)−k, γ =

(a bc d

)∈ Γ0(N) .

Let N > 1 then the action of the involution τN =

(0 −1N 0

)brings Gk(z, χ) to a similar series but

without a constant term:

G′k(z, χ) =

∞∑

n=1

σ′k−1,χ(n)qn ∈Mk(Γ0(N), χ,Q(χ)) ,

where

σ′k−1,χ(n) =

∑

d|n

χ(d)(nd

)k−1

= nk−1∏

p|n

( ordpn∑

i=1

χ(p)−ip−i(k−1)

)

is a finite Euler product.

In the Siegel modular case Γn = Sp2n(Z) ⊃ Γn0 (N), and there is an analogue

Ek(z, χ) =∑

( a b

c d )∈P∩Γ(n)0 (N)rΓ

(n)0 (N)

χ(det d) det(cz + d)−k

of Gk(z, χ). However, for N > 1, the Fourier expansion is known only for the involuted series Ek|τN ,

where τN =

(0n − 1n

N · 1n 0n

)(for N = 1 both series coincide). Here z ∈ Hn is in the Siegel upper half plane:

Hn ={z = tz ∈Mn(C)|Imz > 0

}, and P =

{(a b0 c

)∈ Sp2n(R)

}

is the Siegel parabolic subgroup.

Recall some definitions concerning Siegel modular forms. The symplectic group

Sp2n(R) ={g ∈ GL2n(R)|tg · J2ng = J2n

}(J2n =

(0n −1n

1n 0n

))

acts on the Siegel upper half plane Hn by g(z) = (az + b)(cz + d)−1, where the bloc notation g ={(

a bc d

)∈ Sp2n(Z)|c ≡ 0 mod N

}denote a congruence subgroup of the Siegel modular group Γn =

Sp2n(Z).

17

A Siegel modular form f ∈Mk(Γn0 (N), χ) is a holomorphic function f : Hn → C such that for every

γ =(

a bc c

)∈ Γn

0 (N) one has

f(γ(z)) = χ(det d) det(cz + d)kf(z)

(for n > 1 the regularity at ∞ is automatically satisfied by Koecher).

The Fourier expansion of such f uses the semi-group

Bn = {ξ = tξ ≥ 0|ξ half-integral}, and qBn (a multiplicative semi-group), so that

f(z) =∑

ξ∈Bn

a(ξ)qξ ∈ C[[qBn ]]( a formal q − expansion ∈ Λ[[qBn ]]),

where the symbols

qξ = exp(2πitr(ξz)) =

m∏

i=1

qξii

ii

∏

i<j

q2ξij

ij ⊂ C[[q11, . . . , qmm]][qij , q−1ij ]i,j=1,··· ,m

are used (with qij = exp(2π(√−1zi,j))); they form a multiplicative semi-group so that qξ1 · qξ2 = qξ1+ξ2 ,

and one may consider f as a formal q-expansion. In this way one can introduce Siegel modular forms

over an arbitrary ring A as certain elements of the semi-group algebra A[[qBm ]].

Definition 3.1. LetMn(N,χ,ΛO) denote the ΛO-submodule of ΛO[[qBn ]] generated by all F =∑

ξ∈Bn

a(ξ, T )qξ

such that for almost all k ∈ N we have that

Fk =∑

ξ

a(ξ, (1 + p)k − 1)qξ ∈Mnk (Γn

0 (N)), χω−k) .

Definition 3.2. More generally, let ΛY,O = O[[Y ]] be as in §1, and let P ⊂ C(Y,O) be a family of O-

valued continuous functions on Y , MnP(N,χ,ΛY,O) is the ΛY,O-submodule of ΛY,O[[qBn ]] generated by all

F =∑

ξ∈Bn

aξqξ (aξ ∈ ΛY,O) such that for almost all P ∈ P

FP =∑

ξ∈Bn

(∫

Y

Pdaξ

)qξ =

∑

ξ∈Bn

aξ(P )qξ

is the Fourier expansion of a Siegel modular form of weight k = k(P ), level N = N(P ) and character

χ = χ(P ).

In the same way, one defines MnP(N,χ,LY,O), LY,O = QuotΛY,O and

SnP(N,χ,ΛY,O) ⊂ Sn

P(N,χ,LY,O).

18

Example. Involuted Siegel-Eisenstein series: let k > n+ 1, and let χ be a Dirichlet character modulo N .

Let us define

E = E(k, χ) =∑

γ∈PrΓn0 (N)

χ(det d) det(cz + d)−k ∈Mk(Γ0(N), χ) , (3.1)

where γ =(

a bc d

)∈ Γn

0 (N). Note also that (see [Nag2], p.408)

E(2)4 (z) =1 + 240q11 + 240q22 + 2160q211 + (240q−2

12 + 13440q−112

30240 + 13440q12 + 240q212)q11q22 + 2160q222 + . . .

E(2)6 (z) =1− 504q11 − 504q22 − 16632q211 + (−540q−2

12 + 44352q−112

166320 + 44352q12 − 504q212)q11q22 − 16632q222 + . . . .

Let us also define G+(k, χ) = L+(k, χ)E(k, χ)|τN , τN =

(0n −1n

N · 1n 0n

), where

E|τN = Nkn/2(det Nz)−kE(−(Nz)−1) ,

L+(k, χ) = Γ(k, n)LN(k, χ)

[n/2]∏

i=1

LN(2k − 2i, χ2) , and

Γ(k, n) = ink2−n(k+1)π−nkΓn(k),Γn(s) = πn(n−1)/4n−1∏j=0

Γ(s− (j(2))) . (3.2)

Theorem 3.1 (Siegel, Shimura [Sh83], P. Feit [Fe]). If n is odd, k > n+ 1, N > 1, then

G+(k, χ) =∑

0<ξ∈Bn

b+(ξ, k, χ)qξ, b+(ξ, k, χ) = det ξk−(n+1)/2M(ξ, k, χ) , (3.3)

where M(ξ, k, χ) =∏

b| det(2ξ)

Mq(ξ, χ(ℓ)ℓ−k), Mℓ(ξ, x) ∈ Z[x] (a finite Euler product, e.g. for n = 1

Mℓ(n, x) =ordℓn∑i−0

xi).

In order to construct a family, put G+p (k, χ)

det=

∑ξ>0

p∤det(2ξ)

b+(ξ, k, χ)qξ.

Theorem 3.2. (see [PaSE]) Let χ mod Np be a Dirichlet character, O ⊃ Zp[x] then there exists EλO(χ, i) ∈

ΛO[[qBn ]] such that for all k ≫ 0, k ≡ i mod (p− 1)

EΛO(χ, i)

T=(1+p)k−1= G+

p (k, χ−1ωk) (3.4)

EΛ(χ) is called the Λ-adic Siegel-Eisenstein series.

19

Proof of Theorem 3.2: For any δ ∈ ∆ = Z∗torsp let us introduce the notation

G+p,δ(k, χ) =

∑

ξ>0det ξ≡δ mod p

b+(ξ, k, χ)qξ (3.5)

then

G+p (k, χ−1ωk) =

∑

δ∈∆

∑

ξ>0det ξ≡δ mod p

det ξk−((n+1)/2) ·M(ξ, k, χ−1ωk)qξ

=∑

δ∈∆

∑

ξ>0det ξ≡δ(mod p)

(det ξ)−(n+1)/2 (3.6)

·〈det ξ〉k∏

ℓ| det(2ξ)

Mℓ(ξ, χ−1ωk(ℓ)ℓ−k)qξ

Note that ωk(ℓ)ℓ−k = 〈ℓ〉−k hence the coefficient of qξ in (3.6) is

∑

j

αjβkj (a finite linear combination) where αj ∈ O, βj = (1 + p)γj ∈ Γ.

It follows that we can define EΛO(χ, i) as formal series in ΛO[[qBn ]]:

EΛO(χ, i) =

∑

δ∈∆

δi∑

ξ>0det ξ≡δ mod p

∑

j

αj(1 + T )γj · qξ ∈ ΛO[[qBn ]] (3.7)

such that if we substitute T = (1 + p)k − 1 in (3.7) we obtain

∑

δ∈∆

δi∑

ξ>0det ξ≡δ mod p

∑

j

αj(1 + p)γjk · qξ = G+p (k, χ−1ωk) .

Remarks. a) For any k we can specialize EΛO(χ, i) to T = (1 + p)k − 1:

EΛO(χ, i)

T=(1+p)k−1=∑

δ∈∆

δi−k∑

ξ>0det ξ≡δ

(det ξ)k−((n+1)/2)M(ξ, k, χ−1ωk)qξ

= G+p (k, χ−1ωk)ωi−k ,

where the subscript ωi−k denotes the twist of G+p (k, χ−1ωk) ∈ Mn

k (Γ0(Np), χω−k) with the Dirichlet

character ωi−k.

20

b) Let An = {ξ = tξ|ξ half-integral} be the lattice in Mn(Q) dual to {η ∈ Mn(Z)|tη = η} with respect

to the pairing (ξ, η) 7→ Tr(2πiξη).

Put An,p = An ⊗Z Zp then Theorem 3.2 can be written in a more general form: there exists a p-adic

measure µSiegel−Eisenstein = µSE on the profinite group Y = An,p × Z∗p with values in O[[qBn ]] such that

for each k > n+ 1 and for each Dirichlet character χ : Z∗p → O∗

∫

An,p×Z∗p

det ξk−((n+1)/2)χ(x)xkµSE(ξ, x) = G+p (k, χ−1) .

c) Theorem 3.2 and its generalization b) can be extended also to even n, but in this case the formula

for b+(ξ, k, χ) contains also the factor L(k− (n/2), χωξ), where ωξ is a quadratic character associated to

ξ,ωξ(d) =

((−1)n/2 det(2ξ)

d

). Note that we need to assume χ(−1) = (−1)k, and this additional factor

is interpreted by the Kubota-Leopoldt theorem, see §1.

21

4 Λ-adic Klingen-Eisenstein series

In this section we construct another example of a p-adic measure with values in the vector space of Siegel

modular forms.

Recall the classical case: let f ∈ Sk(Γr) be a Siegel cusp form of weight k for the group Γr = Spr(Z).

For any m≥ r define the Klingen parabolic subgroup

Pm,r = m−r

{(∗ ∗

0 . . . 0 ∗

)

︸︷︷︸m+r

⊂ Sp2m(R)

and put

E(f) = Em,rk (z, f) =

∑

γ∈P m,r∩ΓmrΓm

f(ωr(z))kγ, k > m+ r + 1 (4.1)

where z ∈ Hm, ωr(z) is the upper left r × r corner of z.

Theorem 4.1 (M. Harris, see [Ha81]). E(f) ∈ Q(f)[[qBn ]].

In order to construct a p-adic family of such series we normalize it as follows

E∗k(z, f) = Λ(k)

D(k − r, f)

〈f, f〉 E(z, f) , (4.2)

where Λ(k) is a product of special values of Dirichlet L-functions and Γ-functions, D(s, f) is the standard

zeta function of Andrianov-Kalinin (of degree 2r+1) whose critical values are s = 1+r−k, . . . , k−r, k ≥ 2r.

After this normalization, the following Böcherer’s formula holds

E∗(z, f) =

⟨f(w), Em+r

k (diag{z, w}⟩w

〈f, f〉 (4.3)

in which Em+rk denotes the Siegel-Eisenstein series of degree n = m + r, weight k, for the full Siegel

modular group Sp2n(Z), see [Bö85], [Shi95]. Let us imitate p-adically this formula in the case r = 1,

m = 2, n = m+ r = 3. Consider an ordinary Λ-adic elliptic cusp form f ∈ Sord(χ,Λ) where χ = ωi.

Theorem 4.9 (K. Kitagawa, A. Panchishkin, see [PaSE]). There exists a Λ-adic Siegel modular form

EΛ(χ, f) of degree 3 which is given by a Λ-adic version of Böcherer’s formula (4.3).

Proof: The construction of EΛ(χ, f) uses the following main ingredients:

• The algebraic Petersson product (f, f) ∈ L = Quot Λ (§2).

22

• The algebraic pullback formula (“the restriction to diag{z, w}”): if F ∈ Λ[[qBn ]], F =∑

ξ∈Bn

a(ξ)qξ then

Φm,rn (F )(“= F (diag{z, w})”) =

∑

(ξ1,ξ2)∈Bm×Br

a(ξ1, ξ2)qξ1qξ2 , (4.4)

where

a(ξ1, ξ2) =∑

ξ=“

ξ1 ∗

∗ ξ2

”

∈Bm+r

a(ξ) (this is a finite sum).

• The algebraic kernel linear operator (“integration over w”).

Definition 4.2. a) The Λ-module Mm,r(χ,Λ) is the Λ-submodule of Λ[[qBm×Br ]] generated by all

elements F =∑

(ξ1,ξ2)∈Bm×Br

a(ξ1, ξ2)qξ1qξ2 such that for almost all k ≫ 0

Fk =∑

ξ1,ξ2

κk(a(ξ1, ξ2))qξ1qξ2

is an element of Mm,rk (Np, χω−k).

b) The Λ-module Mm,1−ord(χ,Λ) = eξ2Mm,1(χ,Λ) is defined by applying Hida’s projector e to the

second variable qξ2 (on the formal q-expansions with coefficients in Λ).

Theorem 4.3. Mm,1−ord(χ,Λ) = Mm(χ,Λ)⊗Λ M1−ord(χ,Λ).

Proof of 4.3: uses only the fact that M1−ord(χ,Λ) is a finitely generated Λ-module, see §1.

It follows from Theorem 4.3 that there is a natural pairing

Sord(χ,Λ)×M2,1(χ,Λ)→M2(χ,Λ), (f,G) 7−→ R(f, eξ2G) ,

where eξ2G ∈M2(χ,Λ)⊗M1−ord(χ,Λ), and R denotes the following contraction operator:

R : (f, g ⊗ h) 7−→ g · (f, h) ,

f ∈ Sord(χ,Λ), g ∈M2(χ,Λ), h ∈M1−ord(χ,Λ) .

Note that (f, h) ∈ L = Quot Λ is (Hida’s algebraic scalar product, defined over the field L, see §2),

and (f, h) = (f, h−EisLh) where EisLh is the Eisenstein projection of h (which is defined only over

the field L).

23

The series EΛ(χ, f) of Theorem 4.2 is then given by the Λ-adic Böcherer’s formula:

EΛ(χ, f) =1

(f, f)R(f,Φ2,1

3 E3Λ(χ)) . (4.5)

Note that a similar construction works in order to construct families of Klingen-Eisenstein series, at-

tached to families of cusp forms of positive slope. Their Fourier coefficients should interpolate symmetric

squares of these cusp forms in families.

24

5 Λ-adic theta functions with spherical polynomials

In this section we describe certain p-adic families of Siegel modular forms whose parameter space is

Y = Xp = X ⊗ Zp where X = Zmn is a lattice. Let A ∈ Mm×m(Q) be a positive definite half-integral

symmetric matrix which determines a quadratic form QA(x) = txAxdef=A[x] for x ∈ X .

Theta-distribution attached to A (see [Att], [Hi85]) is an element θA ∈ Distr(Y,O[[Bn]]) defined by

θA(U) =∑

x∈U∩X

qA[x] U a compact open subset of Xp (5.1)

In other words, θA(U) is a partial theta-series given by congruence conditions defining U , (see also [Hi85]

where the elliptic modular case was treated).

Let ΛX = Distr(Xp,O) = ΛY,O be the generalized Iwasawa algebra (which is isomorphic to O[[T11, . . . , Tmn]]:

for every µ ∈ Distr(Xp,O) the corresponding power series is given by the Amice transform:

Aµ(T11, . . . , Tmn) =

∫

Zmnp

(1 + T11)x11 · · · (1 + Tmn)xmnµ(x), (5.2)

where (1 + T )x =∑

j ≥ 0

(xj

)T j,

(x

j

)=x(x − 1) · · · (x− j + 1)

j!.

One observes directly from definition (5.1) that

θA =∑

x∈Zmn

δx · qA[x] ∈ ΛX [[qBn ]] .(5.3)

Using the Amice transform, we then rewrite (5.3) in the form of a power series

θA(T11, . . . , Tmn; q) =

∫

Zmnp

(1 + T11)x11 · · · (1 + Tmn)xmndθA(x) =

∞∑

k11,...,kmn=0

T k1111 · · ·T kmn

mn

∫

Zmnp

(x11

k11

)· · ·(xmn

kmn

)dθA(x) = (5.4)

∑

ξ∈Bn

∞∑

k11,...,kmn=0

∑

x∈Zmn

A[x]=ξ

T k1111 · · ·T kmn

mn

(x11

k11

)· · ·(xmn

kmn

)qξ =

∑

x∈Zmn

(1 + T11)x11 · · · (1 + Tmn)xmnqA[x].

Let us consider the family P = {P : X → Z} of homogeneous A-spherical polynomials which take

integral values on the lattice X = Zmn. Then we may view P as a Zp-valued continuous function on

Xp = Zmnp and ∫

Xp

PdθA(x) = θA,P =∑

x∈Z

P (x)qA[x] (5.5)

25

is a Siegel modular form of weight m2 + deg P (we assume that m is even).

The modular forms (5.5) can be obtained as certain specialization of the power series (5.4), and we

obtain a p-adic family of theta series with spherical polynomials.

Note that

θA,P =∑

x∈Zmn

P (x)(1 + T11)x11 · · · (1 + Tmn)xmnqA[x]

T11,...,Tmn=0

= ∂P θA(T11, . . . , Tmn; q)T11,...,Tmn=0

(5.6)

where

∂P = P

((1 + T11)

∂

∂T11, . . . , (1 + Tmn)

∂

∂Tmn

)

is a differential operator such that

∂P (1 + T11)x11 · · · (1 + Tmn)xmn = P (x)(1 + T11)

x11 · · · (1 + Tmn)xmn (5.7)

The map F 7→ ∂PFT11,...,Tmn=0

defines the following homomorphisms κP : ΛX → Zp, and

κP : ΛX [[qBn ]]→ Zp[[qBn ]].

Let us consider the Laplace transform

Lµ(T11 . . . , Tmn) =

∞∑

k11,...,kmn=0

T k1111 · · ·T kmn

mn

k11! · · · kmn!

∫

Zmnp

xk1111 · · ·xkmn

mn dµ(x)

of a p-adic measure µ ∈ Distr(Zmnp ,O) with values in a complete ring O. Then by definition of the Amice

transform we have

Lµ(T11, . . . , Tmn) = Aµ(eT11 − 1, . . . , eTm,n − 1),

Lµ(T11, . . . , Tmn) =

∫

Zmnp

eT11x11+···+Tmnxmndµ(x) (5.8)

Note that the coefficients of the power series (5.8) may contain denominators k11! · · · kmn! so that

Lµ(T11, . . . , Tmn) ∈ KO[[T11, . . . , Tmn]]

where KO = Quot O.

If

P =∑

k11,...,kmn

bk11,...,kmnxk11

11 · · ·xkmnmn

26

is the spherical polynomial in (5.5), then the integral

∫

Zmnp

PdθA(x) =∑

k11,...,kmn

bk11,...,kmn

∫

Zmnp

xk1111 · · ·xkmn

mn dθA

is related to the Laplace transform by a more simple formula than (5.6), namely

LθA(T11, . . . , Tmn) =

∑

x

eT11x11+···+TmnxmnqA[x],

θA,P = P

(∂

∂T11, . . . ,

∂

∂Tmn

)Lθ

T11=...=Tmn=0(5.9)

A nice arithmetical example of a p-adic family of Hecke cusp eigenforms which are theta functions with

spherical polynomials come from Hecke grössencharacters of an imaginary quadratic fieldK = Q(√−D

).

Recall that such a χ of conductor (dividing) a given integral ideal m of K is by definition a homo-

morphism

{prime-to-m fractional ideals of K} → Q×

such that for appropriate integers n = n(λ)

χ((α)) =∏

λ

λ(α))n(λ) = σ(α)n(σ)σ(α)n(σ) if α ≡ 1 mod×m

where λ : K →֒ Q, λ = σ, σ, σ : K →֒ Q is a fixed embedding, and σ is its composition with the

nontrivial Galois automorphism of K.

In order to define L(s, χ) one uses a fixed complex embedding of Q. If n(σ) > 0 and n(σ) = 0 then

there exists an elliptic cups eigenform fχ ∈ Sk(Γ0(N(m)), ψ) of weight k = n(σ) + 1 with a Dirichlet

character ψ (depending on χ) such that L(s, χ) = L(s, fχ) (the Mellin transform of fχ). Moreover fχ

is a linear combination of binary theta-series with spherical polynomials defined by the values of the

grössencharacters χ (see [Bl86]).

The function χ 7→ fχ provides us with a p-adic family of Hecke cusp eigenforms (see [Hi85], [Ka78],).

It would be very interesting to extend this example to all CM-fields K in the spirit of [Ka78], and

then to interpret the p-adic L-functions constructed by N.M.Katz in terms of these p-adic families (of

Hilbert modular forms). In this case the prameter group Y = Gal(Kab/K) coincides essentially with the

27

idèle class group A∗K/K

∗, and the p-adic L-functions are Cp-analytic on the corresponding Cp-analytic

Lie group Xp,K = Homcont(A∗K/K

∗,C∗p).

28

6 p-adic families of automorphic representations

We discuss in this section a general (conjectured) definition of a p-adic family of automorphic representa-

tions (or of a Λ-adic automorphic form). We shall view the Iwasawa algebra ΛO = O[[T ]] as the algebra of

all O-valued distributions on Zp (with the additive convolution as a multiplication): ΛO = Distr(Zp,O).

Let VQ ⊂ C(G(AF )) be a certain Q-vector space of some (complex-valued) continuous functions on

the adelic group G(AF ) over a number field F where G is a semi-simple F -algebraic group. We assume

that VQ is endowed with an integral structure VZ so that VQ = VZ⊗Z Q. Put Vp = VZ⊗̂Zp (the completed

tensor product). Let us define Dp(Vp) = Distr(Zp, Vp) (this is a module over Λ = Dp(Zp)).

Definition 6.1. A p-adic family of automorphic representations on G is a p-adic measure ϕ ∈ Dp(Vp) such

that for almost all positive integers k we have that the integral∫

Zp

xkdϕ = ϕk ∈ Vp belongs to VZ and

the function ϕk generates an automorphic representation πk of the group G(AF ). We call ϕ an Λ-adic

automorphic form on G(AF ).

Definition 6.2. Let AFG(Λ) denote the Λ-submodule of Dp(Vp) generated by all such elements ϕ. An

element ϕ is called an eigenform if the representations πk are all irreducible.

Notation: AFG(L) = AF (Λ)⊗λ

L, L = Quot Λ.

In the same way, one can define p-adic families of automorphic forms parametrized by a subset

P ⊂ C(Y,O) of continuous O-valued functions on a profinite parameter space Y : ΛY,O = O[[Y ]] (the

completed group ring), DY (VO) = Distr(Y, VO), VO = VZ⊗̂O, so that ΛY,O = DY (O) = Distr(Y,O) and

DY (VO) is a ΛY,O-module.

Definition 6.3. A p-adic family of automorphic representations on G parametrized by a set P ⊂ C(Y,O) is

a p-adic measure ϕ ∈ DY (VO) such that for almost all P ∈ P we have that the integral∫

Y

Pdϕ = ϕP ∈ VO

belongs to VZ, and the function ϕP generates an automorphic representation πP of the group G(AF ).

We call ϕ a ΛY,O-adic automorphic form with respect to the family P.

Definition 6.4. Let AFG(ΛY,O) denote the ΛY,O-submodule of DY (VO) generated by all such elements

ϕ ∈ DY (VO). An element ϕ is called an eigenform if the representation πP are all irreducible.

29

Notation: AFG(LY,O) = AF (ΛY,O)⊗ΛY,OLY,O, LY,O = Quot ΛY,O.

A natural example of such a vector space V for the group G = GL2 over F = Q comes from holomorphic

functions f =∑

n ≥ 0ane

2πinz on the upper half plane H , having rational Fourier coefficients an ∈ Q with

bounded denominators, i.e. for which there exists a positive integer N = N(f) such that N · an ∈ Z;

then VZ comes from f =∑

n ≥ 0ane

2πinz (an ∈ Z) and Vp becomes Zp[[q]].

However, there are other ways to attach such a vector space V to G by considering cohomology

groups of the corresponding locally-symmetric spaces and automorphic forms ϕ on G(AF ) represented

by rational cohomology classes (see, for example, [Ko-Za]).

We hope that one could find in this way a general construction of p-adic automorphic L-functions

Lπ,r,p to (complex-valued) Langlands L-functions L(s, π, r) where π is an automorphic representation of

G(AF ) and r : LG(C)→ GLm(C) a finite dimensional representation of the Langlands group LG(C). In

general Lπ,r,p should belong to LY,O, and they should also interpolate (normalized) special values

P 7−→ L∗(sP , πP , r) (P ∈ P)

in a family P ⊂ C(Y,O).

If such a family {πP } comes from a Λ-adic cuspidal automorphic eigenform ϕ ∈ AFG(LY m,O), then

we hope that there exists a LY,O-linear form ℓ = ℓG,r,

ℓG,r : AFG(LY mO)→ LY,O

representing its p-adic L-function: Lπ,r,p = ℓG,r(ϕ), where π = πP0 is a member of the family {πP }parametrized by P. Note that P can be very large, and include the cyclotomic part, the anti-cyclotomic

part, etc. An important example in the case of the group G = GL1 over a number field is given by the

family ChA0(A∗F ) of all grössencharacters of type A0 of the idèle class group A∗

F /F∗ (in the sense of Weil

[We56]). In this case

ChA0(A∗F ) ⊂ Homcont(A

∗F /F

∗,C∗) and ChA0(A∗F ) ⊂ Homcont(A

∗F /F

∗,Cp)

for all primes p.

For a general semi-simple group G/F , a natural assumption on a family P parametrizing {πP } would

be its invariance under central twists by elements χ ∈ ChA0(A∗F ): {πP } 7−→ {πP ⊗ χ}, see [Ka78].

To conclude, there exist nice constructions of p-adic families of Galois representations attached to

automorphic forms (Λ-adic Galois representations, see [Hi86], [Til-U]) which were important for the work

of Wiles [Wi95]. It would be interesting to formulate a general Λ-adic Langlands conjecture relating Λ-

adic automorphic forms and Λ-adic Galois representations.

30

7 Application: a solution of the problem of Coleman-Mazur

(in [PaTV] by A.P., Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent.

Math. v. 154, N3 (2003), pp. 551 - 615).

For a prime number p ≥ 5, consider a primitive cusp eigenform f = fk of weight k ≥ 2, f =∑∞

n=1 anqn, and consider a family of cusp eigenforms fk′ of weight k′ ≥ 2, k′ 7→ fk′ =

∑∞n=1 an(k′)qn,

containing f for k′ = k, such that the Fourier coefficients an(k′) are given by certain p-adic analytic

functions k′ 7→ an(k′) for (n, p) = 1, and let αp(k′) be a Satake p-parameter of fk′ .

In "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following problem:

Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ =

ordp(αp(k′)) > 0, to construct a two variable p-adic L-function interpolating on k′ the Amice-Vélu

p-adic L-functions Lp(fk′ ).

A solution (2003) is described using the Rankin-Selberg method and the theory of p-adic integration

with values in a p-adic algebra A.

Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures

come from Eisenstein distributions with values in certain Banach A-modules M† = M†(N ; A) of families

of overconvergent forms over A.

Another approach, based on overconvergent families of modular symbols, was developed by Glenn

Stevens. Applications of these results to the p-adic Birch and Swinnerton-Dyer conjecture were discussed

by P.Colmez (Bourbaki talk, June 2003, [Colm03]).

The Tate field Cp

Fix a prime p, and let Cp = Q̂p be the Tate field(the completion of the field of p-adic numbers)

We fix an embedding ip : Q→ Cp,and view algebraic numbers asp-adic numbers via ip.

A primitive cusp eigenform f

f = fk =∑

n≥1

anqn ∈ Sk(Γ0(N), ψ),

(where q = e(z) = exp(2πiz), Im(z) > 0)

A primitive cusp eigenform f = fk

of weight k ≥ 2 for Γ0(N)with a Dirichlet character ψ (mod N).

31

The special values of the L-function attached to f at s = 1, · · · , k − 1:

Lf(s, χ) =∑

n≥1

χ(n)ann−s,

(χ are Dirichlet characters)

where 1− apX + ψ(p)pk−1X2 = (1− αX)(1− α′X)is the Hecke polynomialα and α′ are called the Satake parameters of f

Periods of f

Following a known theorem of Manin [Ma73], there exist two non-zero complex constants c+(f), c−(f) ∈C× (the periods of f) such that for all s = 1, · · · , k − 1 and for all Dirichlet characters χ of fixed parity,

(−1)k−sχ(−1) = ±1, the normalized special values are algebraic numbers:

L∗f (s, χ) =

(2iπ)−sΓ(s)Lf (s, χ)

c±(f)∈ Q. (7.1)

A family of slope σ > 0 of cusp eigenforms fk′ of weight k′ ≥ 2 containing f

k′ 7→ fk′ =∞∑

n=1

an(k′)qn ∈ Q[[q]]

1) the Fourier coefficients an(k′) of fk′

and the Satake p-parameter αp(k′) are given by certain

p-adic analytic functions k′ 7→ an(k′) for (n, p) = 12) the slope is constant and positive:ord(αp(k

′)) = σ > 0

A model example of a p-adic family (not cusp and σ = 0): Eisenstein series

an =∑

d|n

dk′−1, fk′ = Ek′

the fk′ the Fourier coefficients an(k′) of Ek′

and one of the Satake p-parameters αp(k′) = 1

ordp(αp(k′)) = ordp(1) = 0

The existence of families of slope σ > 0: R.Coleman, [CoPB]

He gave an example with p = 7, f = ∆, k = 12

a7 = τ(7) = −7 · 2392, σ = 1, and

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

The Problem, see [Co-Ma] R. Coleman, B. Mazur, The eigencurve. Galois representationsin arithmetic algebraic geometry, (Durham, 1996), London Math. Soc. Lecture Note Ser.,254, at p.6

Given a p-adic analytic family k′ 7→ fk′ =

∞∑

n=1

an(k′)qn ∈ Q[[q]] of positive slope σ > 0, to construct a

two-variable p-adic L-function interpolating L∗fk′

(s, χ) on (s, k′).

32

Known cases:

• One-variable case(k = k′ is fixed, σ > 0),

treated in [Am-Ve] by Y. Amice, J. Vélu,in [Vi76] by M.M. Višik, and in[MTT] by B. Mazur; J. Tate; J. Teitelbaum

• σ = 0 (H.Hida)("ordinary families")

(see in [Hi93] H. Hida, Elementary theory of L-functionsand Eisenstein series, London Mathematical SocietyStudent Texts. 26, Cambridge University Press, 1993

•

Special values of L-functions attached tofamilies fk of Yu.I. Manin and M. M.Vishik, [Ma-Vi]

fk =∑

a⊂OK

λk−1(a)qNa

and of N.M.Katz, [Ka78]),which are are certain ordinary families

they correspond to powers of agrössen-character λ of an imaginaryquadratic field K at a splitting prime p,(resp. to grössencharacters of type A0

of the idèle class group A∗K/K

∗

(in the sense of Weil [We56],)of a CM-field K.

Motivation:

comes from the conjecture of Birch and Swinnerton-Dyer, see in [Colm03] , Colmez, P.: La conjecture

de Birch et Swinnerton-Dyer p-adique. Séminaire Bourbaki. [Exposé No.919] (Juin 2003). For a cusp

eigenform f = f2, corresponding to an elliptic curve E by Wiles [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 formula needsthe existence of our two variable L-function!

Our method

is a combination of the Rankin-Selberg method with the theory of p-adic integration with values in p-adic

Banach algebras A and the spectral theory of Atkin’s U -operator: U = Up : A[[q]]→ A[[q]] defined by:

U

∑

n≥1

anqn

=∑

n≥1

apnqn ∈ A[[q]], (Uf)(z) :=

1

p

p−1∑

m=0

f

(z +m

p

).

33

Here A = A(B) is a certain p-adic Banach algebra of functions on an open analytic subspace B on the

weight space X = Homcont(Y,C∗p). This is an analytic space over Cp, which consists of all continuous

characters of a certain profinite group Y over Z∗p.

The classical analogue of the weight space is the complex plane

C = Homcont(R∗+,C

∗), s 7→ (y 7→ ys).

The weights k′ correspond to certain points in the neighborhood B of the given weight k.

Any series f =∑

n≥1 anqn ∈ A[[q] produces a family of q-expansions

{fk′ = evk′(f) =

∑

n≥1

evk′(an)qn ∈ Cp[[q]]}, which can be classical modular forms in Q[[q]].

• We construct first an analytic function Lµ : X → A = A(B) as the Mellin transform

Lµ(x) =

∫

Y

xdµ

of a certain measure µ on our profinite group Y with values in A.

• For each s ∈ B, there is the evaluation homomorphism evs : A(B) → Cp; we obtain Lµ(x, s) by

evaluation of an A-valued integral:

Lµ(x, s) = Lµ(x)(s) = evs

(∫

Y

xdµ

)(x ∈ X, Lµ(x) ∈ A).

This gives a p-adic analytic L-function in two variables (x, s) ∈ X ×B ⊂ X ×X .

• We check an equality relating the algebraic numbers L∗fk′

(s, χ) (s = 1, · · · , k′ − 1) with the values

Lµ(x, k′) at certain arithmetic characters x ∈ X .

Another approach (Glenn Stevens, unpublished)

uses overconvergent families of modular symbols, see [Ste]. As noted by Stevens, it yields a formula for

the derivative at s = k − 1 of the p-adic L-function of fk′ .

34

8 General L-functions of two variables and admissible measures

Let us use the theory of p-adic integration with values in p-adic Banach algebras A. A useful notion of

an h-admissible measure was introduced by Y. Amice, J. Vélu, cf. [Am-Ve] (see also [MTT], [Vi76]):

Definition 8.0.1 a) For h ∈ N, h ≥ 1 let Ph(Y,A) denote the A-module of locally polynomial functions of

degree < h of the variable yp : Y → Z×p →֒ A×; in particular,

P1(Y,A) = C

loc−const(Y,A)

(the A-submodule of locally constant functions). Let also denote Cloc−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).

b) Let V be a normed A-module with the norm | · |p,V . For a given positive integer h an h-admissible

measure on Y with values in V is an A-module homomorphism

Φ̃ : Ph(Y,A)→ V

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

∫

a+(Npv)

(yp − ap)h′

dΦ̃

∣∣∣∣∣p,V

= o(p−v(h′−h)) (8.1)

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

The condition (8.1) allows one to integrate all locally-analytic functions: there exists a unique extension

of Φ̃ to Cloc−an(Y,A)→ V (via the embedding Ph(Y,A) ⊂ Cloc−an(Y,A)). The integral is defined using

generalized Riemann sums : take the beginning of the Taylor expansion of a locally-analytic function

φ ∈ Cloc−an(Y,A) (of order h− 1) instead of just values of a function φ.

By Definition 8.0.1, an h-admissible measure on a profinite group Y with values in A is an A-linear

homomorphism

µ̃ : Ph(Y,A)→ A,

given by a sequence {µj} of distributions on Y , in such a way that for j = 0, 1, · · · , h − 1 and for all

compact open subsets U ⊂ Y one has

∫

U

yjpdµ̃ = µj(U), (8.2)

35

with the following growth condition (8.1): for t = 0, 1, · · · , h− 1

∣∣∣∣∣

∫

a+(Npv)

(yp − ap)tdµ̃

∣∣∣∣∣p

(8.3)

=

∣∣∣∣∣∣

t∑

j=0

(t

j

)(−ap)

t−jµj(a+ (Npv))

∣∣∣∣∣∣p

= o(pv(h−t)) for v →∞.

8.1 A general construction of L-functions of two variables

Let us use the theory of Atkin’s U -operator U = Up on formal q-expansions

U : A[[q]]→ A[[q]],A[[q]][R]→ A[[q]][R],

which correspond to the classical complex notation

q = exp(2πiz), z = x+ iy, R =1

4πy=

1

−2πi(z − z̄) :

U

∑

n≥1

anqn

=∑

n≥1

apnqn ∈ A[[q]], (Uf)(z) :=

1

p

p−1∑

m=0

f

(z +m

p

),

U

∑

n,m∈N

am,nRmqn

=∑

n,m∈N

am,pn(pR)mqn ∈ A[[q]][R]

Here A = A(B) is a certain p-adic Banach algebra of functions on an open analytic subspace B on the

weight space X = Homcont(Y,C∗p). This is an analytic space over Cp, which consists of all continuous

characters of a certain profinite group Y over Z∗p.

The classical analogue of the weight space is the complex plane

C = Homcont(R∗+,C

∗), s 7→ (y 7→ ys).

The weights k′ correspond to certain points in the neighborhood B of the given weight k.

Any series f =∑

n≥1 anqn ∈ A[[q] produces a family of q-expansions

{fk′ = evk′(f) =

∑

n≥1

evk′(an)qn ∈ Cp[[q]]}, which can be classical modular forms in Q[[q]].

Consider a two parameter family of algebraic special values of some L-function D∗fk′

(s, χ):

(k′, j) 7−→ D∗fk′

(r∗ + j, χ), where j = 0, · · · , tk′

36

for a family {fk′}k′ (for example, the symmetric squares D∗fk′

(s, χ) = L(Sym2(fk′), χ, s)).

Let us assume that D∗fk′

(r∗ + j, χ) = ℓα(k′)(Ψk′,j(χ)) (j = 0, · · · , tk′) for some C∞-modular forms

Ψk′,j(χ) ∈M∞(Npm, ψ,C) and for a C-linear form

ℓα(k′) : M∞(Npm, ψ,C)→ C such that ℓα(k′)(Uh) = α(k′)ℓα(k′)(Uh) and ℓα(k′)(f0k′) = 1.

We require that there exists w0 ∈ N which regularizes the C∞-modular forms in such a way that

Φk′,j = α−w0Uw0(Ψk′,j) belong to the space

Mr,k′(Npm, ψ,Q) ⊂ Q[[q]][R]

of nearly holomorphic modular forms over Q. Such regularization does not change the special values in

question:

D∗fk′

(r∗ + j, χ) = ℓα(k′)(Ψk′,j(χ)) = ℓα(k′)(Φk′,j(χ)),

Assume also that for all (k′, j) with j = 0, · · · , tk′ and for all sufficiently large k′ one has:

Φk′,j(χ)) = evk′(Φj(χ)) for some algebraic families Φj ∈M(N,ψ,A) ⊂ A[[q]][R].

• Then one can construct an analytic function Lµ : X → A = A(B) as the Mellin transform

Lµ(x) =

∫

Y

xdµ

of a certain h-admissible measure µ on a profinite group Y with values in A.

• For each s ∈ B, there is the evaluation homomorphism evs : A(B)→ Cp; we obtain the two variable

p-adic L-function Lµ(x, s) using the evaluation of the A-valued integral:

Lµ(x, s) = Lµ(x)(s) = evs

(∫

Y

xdµ

)(x ∈ X, s ∈ B, Lµ(x) ∈ A).

This gives a p-adic analytic L-function in two variables (x, s) ∈ X ×B ⊂ X ×X .

• We check an equality relating the algebraic numbers D∗fk′

(r∗ + j, χ) (s = 1, · · · , k′ − 1) with the

values Lµ(x, k′) at certain arithmetic characters x ∈ X .

8.2 How to construct A-valued distributions {µj} from the algebraic special

values D∗

fk′(r∗ + j, χ)

We wish to satisfy the equality of the type

evk′

(∫

Y

χ(y) yjp dµ

)= ip

(D

∗fk′

(r∗ + j, χ),)

(j = 0, · · · , tk′).

37

We construct the distributions µj = µf,α,j from three more simple objects:

µf,α,j = ℓα(πα(Φj)), (j = 0, 1, · · · , tk′),

in such a way that

evk′(ℓα(πα(Φj)(χ))) = evk′(ℓα(Φj)(χ))

= ℓα(k′)(Φk′,j(χ)) = ℓα(k′)(Ψk′,j(χ)) = D∗fk′

(r∗ + j, χ).

• Φj is a sequence of modular distributions on Y with values in an A-module M = M(N,ψ; A) of

overconvergent families of modular forms (it has infinite rank)

MN (ψ; A) :=⋃

v≥0

M(Npv, ψ; A), where M(Npv, ψ; A) = M†r,k(Γ0(Np

v), ψ; A)

such that evk′(Φj) = Φk′,j = α−w0Uw0(Ψk′,j), and

Φj(χ) are q-expansions of certain classical modular forms in A[[q]] or A[[q, q̄]][R1/2, R−1/2])

• πα is the canonical projector over the characteristic A-submodule Mα = M

α(A) of Atkin’s operator

U(∑

n,m∈N am,nRmqn

)=∑

n,m∈N am,pn(pR)mqn.

(Key point: the A-module Mα(A) is locally free of finite rank)

• ℓα ∈ HomA(Mα,A) is a A-linear form such that ℓα(Uh) = αℓα(Uh), and normalized by the equality

ℓα(k′)(f0k′ ) = 1 for the eigenfunctions f0

k′ of U , Uf0k′ = α(k′)f0

k′ .

Theorem 8.2.2 (Criterion of admissibility) Let 0 < |α|p < 1 and h = [ordpα] + 1. Suppose that there exists

a positive integer κ such that the following conditions are satisfied: for all j = 0, 1, · · ·,κh−1 and v ≥ 1,

Φj(a+ (Npv)) ∈M(Npκv, ψ) (the level condition) (8.4)

and the following estimate holds: for all w ≥ max(κv, 1) and for all t = 0, 1, · · ·,κh− 1

Uwt∑

j=0

(t

j

)(−ap)

t−jΦj(a+ (Npv)) ≡ 0 mod p−vt(the divisibility condition) (8.5)

Then the linear form

Φ̃α : Phκ(Y,Q)→M

α ⊂M (8.6)

given by Φ̃α(δa+(Npv)yjp) := πα(Φj(a+ (Npv))

(=

∫

a+(Npv)

yjpdΦ̃

α

)

(for all j = 0, 1, · · ·, hκ − 1), is an hκ-admissible measure.

38

Proof of Theorem 8.2.2 uses the commutative diagramm:

M(Npv+1, ψ; A)πα,v−→ M

α(Npv+1, ψ; A)

Uvy

y≀ Uv

M(Np, ψ; A) −→πα,0

Mα(Np, ψ; A) = Mα(Npv+1, ψ; A)

(8.7)

The existence of the projectors πα,v comes from Coleman’s Theorem A.4.3 [CoPB].

On the right: U acts on the locally free A-module Mα(Npv+1,A) via the matrix:

α · · · · · · ∗0 α · · · ∗0 0

. . . · · ·0 0 · · · α

and α ∈ A

× =⇒ this is an isomorphism over Frac(A),

and one controls the denominators of the modular forms of all levels v by the relation:

πα,v(h) = U−vπα,0(Uvh) =: πα(h) (8.8)

The equality (8.8) can be used as the definition of πα. The growth condition (8.1) for πα(Φj) is deduced

from the congruences (8.5) using the relation (8.8) between modular forms.

8.3 Some advantages of the new p-adic method

The construction can be splitted in several independent steps:

1) Construction of distributions Φj (on a profinite or adelic space Y like Y = 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 rank over some algebra A).

2) Application of a canonical projector of type πα onto a finite dimensional subspace Mα(ψ) of Mα(ψ)

(or over locally free A-module of finite rank over some algebra A):

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

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

3) One proves the admissibility criterium 8.4 saying that the sequence πα(Φj) of distributions with

values in Mα(ψ) determines an h-admissible measure Φ̃ with values in this finite dimensional space

for a suitable h (determined by the slope ordp(α)).

4) Application of a linear form ℓ of type g 7→ 〈f0, πα(g)〉/〈f, f〉 produces distributions µj = ℓ(πα(Φj)),

and (automatically ) an admissible measure: the growth condition is automatically satisfied starting

from congruences between modular forms πα(Φj).

39

5) One shows that certain integrals µj(χ) of the distributions µj coincide with certain L-values; however,

these integrals are not necessary for the construction of measures (already done at stage 4).

6) One shows a resultat on uniqueness for the constructed h-admissibles measures: they are determined

by many of their integrals over Dirichlet characters (not all), for example, only over Dirichlet

characters with sufficiently large conductor (this stage is not necessary, but it is nice to have

uniqueness of in the construction), see [JoH04].

7) If we are lucky, we can prove a functional equation for the constructed measure µ (using the uniqueness

in 6), and using a functional equation for the L-values (over complex numbers, computed at stage

5), for example, for Dirichlet characters with sufficiently large conductor (again, this stage is not

necessary, but it is nice to have a functional equation).

This strategy is applicable in various cases, cf. [PaBdx], [Puy], [Go02].

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