ON CONGRUENCES BETWEEN MODULAR FORMS

RICHARD LAWRENCE TAYLOR

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY.

RECOMMENDED FOR ACCEPTANCE

BY THE DEPARTMENT OF

MATHEMATICS

JUNE 1988

c©Copyright by Richard Lawrence Taylor 1988

All Rights Reserved

To John and Mary

Abstract

An elementary lemma on group cohomology is proved. Applied to certain arithmetic

subgroups of real lie groups this leads to a general method of establishing congruences

between modular forms of different weights. We apply this to establish the existence of

certain p-adic families (Hida families) of Siegel modular forms and of modular forms for

GL2 over an imaginary quadratic field. We also use these methods to show that the exis-

tence of certain Galois representations one expects to be attached to Siegel modular forms

corresponding to holomorphic discrete series would imply their existence for Siegel modular

forms corresponding to limit of holomorphic discrete series.

1

Contents

Abstract 1

Introduction 4

1 A Group Cohomological Lemma 7

1.1 The Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Some Congruences between Siegel Modular Forms 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Review of Siegel Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Relation to Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Some Lemmas on Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 p-adic Families of Siegel Modular Forms 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Review of Siegel Modular Forms II . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Some Lemmas on Eisenstein and Theta Series . . . . . . . . . . . . . . . . . 64

3.4 The General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 Conjectural Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2

4 Modular Forms over an Imaginary Quadratic Field 96

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Review of Cohomology Groups and Automorphic Forms . . . . . . . . . . . 99

4.3 Cohomology of the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4 Change of Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 Change of Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.6 Cyclotomic Hida Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.7 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3

Introduction

This work is concerned with certain methods of Hida for producing congruences between

modular forms. We generalise some of these methods and apply them to the problem of

constructing Galois representations attached to modular forms.

Hida in a series of papers (see [Hi1] and the references cited therein) has constructed

certain p-adic families of elliptic modular forms. To describe these results fix an odd prime

p, an integer N and a Dirichlet character χ defined modulo Np. Fix Qac ⊂ Qacp and

Qac ⊂ C. Let O be the integers of a finite extension of Qp in which χ is valued and let

Λ = O[[T ]]. By a Λ-adic form we mean a formal power series∑∞

n=0 anqn with an ∈ Λ such

that for all pairs (k, α) of an integer k ≥ 2 and a character α : (1 + pZ)/(1 + prZ)→ Qac×,∑∞

n=0 an(α(1 + p)(1 + p)k − 1)qn is the Fourier expansion of an elliptic modular form of

weight k, level Npr and character χω−kα, where ω denotes the Teichmuller character. If

Λ′ is a finite extension of Λ we define the space of Λ′-adic forms to be the space of Λ-

adic forms tensored with Λ′. One can define an action of the Hecke operators on Λ-adic

forms compatible with specialisation. One can also define a notion of ordinary modular

forms, both at the finite and at the Λ-adic level. Roughly speaking the space of ordinary

forms over a p-adic ring of integers, or over Λ, is the largest space on which the action of

Up = [Γ0(Np?)

1 0

0 p

Γ0(Np

?)] is invertible, and its complement is that on which Up

is p-adically nilpotent. Hida’s main results are that the space of Λ-adic ordinary forms of

given level N and character χ is a finite torsion free Λ-module, and moreover any ordinary

eigenform of the Hecke algebra of weight k, level Npr and character χω−kα (with α as

4

above) can be lifted to an ordinary Λ′-adic eigenform of the Hecke algebra (for some finite

Λ′/Λ). Thus there are very systematic families of congruences between ordinary eigenforms.

This work has been generalised by Hida ([Hi2]) and Wiles ([Wi]) to Hilbert modular

forms. In this thesis we shall give generalisations to Siegel modular forms (in chapter 3)

and to modular forms over imaginary quadratic fields (in chapter 4). Our central technique

is a method of comparing modular forms of different weights via group cohomology. This

seems to be a very general method and we give an exposition of it in chapter 1. It is a

generalisation of techniques of Shimura ([Sh1]) and Hida ([Hi1]). It is based on the fact

that if G is an algebraic group, P a parabolic subgroup, M1 and M2 G(Z)-modules and Γ

a congruence subgroup of G(Z) such that Γ mod pr is contained in P (Z/prZ) then there

may be non-trivial Γ-morphisms between M1 ⊗ (Z/prZ) and M2 ⊗ (Z/prZ) which allow us

to compare the Γ-cohomology of the modules M1 and M2.

The easiest type of deduction to draw from this method is that the dimension of the space

of ordinary modular forms for a given group Γ is bounded independently of the weight as the

weight varies over some infinite set. This together with a suitable modular form congruent

to one modulo p is enough to produce a lot of congruences between ordinary eigenforms of

different weights. This is carried out in chapter 2 for Siegel modular forms. In the case of

GSp4 it is applied to the problem of associating Galois representations to modular forms.

Specifically to eigenforms of “weight (n1, n2)” with 2 ≤ n1 ≤ n2 one expects to be able to

associate certain four dimensional Galois representations. If 3 ≤ n1 then one hopes to be

able to find these representations in the cohomology of certain Shimura varieties. However

for the case n1 = 2 no such method is expected to exist. We show how the result for

ordinary eigenforms of weight (2, n2) (n2 ≥ 2) would follow from the result for forms of

weight 3 ≤ n1 ≤ n2.In chapter 3 we organise these congruences into “Hida families”. However we restrict to

the case of parallel weight (k, ..., k) and even genus. We follow the method of Wiles ([Wi])

based on finding enough Λ-adic forms by writing down Λ-adic Eisenstein series, multiplying

these by suitable modular forms and spectrally decomposing the result. This requires that

one already has suitable bounds on the dimension of various spaces of ordinary modular

5

forms involved. This again follows from our results in chapter 1. In the case of Siegel

modular forms the calculations required for this method become very messy.

Finally in chapter 4 we consider the case of imaginary quadratic fields. Here we are not

able to multiply together modular forms, so we work exclusively with the corresponding

cohomology groups. Our results are not as sharp as we would like due to torsion in the

homology groups. As a byproduct of our method we can in fact exhibit torsion in the

homology of certain sheaves (of “non-parallel weight”, so that the torsion free part of the

cuspidal part of the first homology vanishes) on quotients of hyperbolic 3-space by certain

discrete groups.

It is a pleasure to acknowledge the influence of the work of Hida [Hi1] and that of Wiles

[Wi] on this thesis. I have also enjoyed and benefited from many discussions with Fred

Diamond and Michael Larsen. Finally I would like to express my great gratitude to my

advisor Andrew Wiles for his constant help and encouragement.

This work was partially supported by a Sloane Foundation Doctoral Dissertation Fel-

lowship.

6

Chapter 1

A Group Cohomological Lemma

1.1 The Lemma

We shall make repeated use of a certain argument to compare the group cohomology of

different modules, and hence certain spaces of modular forms of different “weight”. We

shall describe this method abstractly as it seems to be applicable rather generally. It is an

extension of some ideas of Shimura ([Sh2]) and of Hida (see [Hi1]).

If ∆ is a semi-group, Γ a subgroup and M a ∆-module then the cohomology groups

H•(Γ,M) may be considered as the image of M under the right derived functors of the

fixed point functor N 7→ NΓ from ∆-modules to abelian groups. If Γ1 and Γ2 are two

subgroups of ∆ and if g ∈ ∆ is such that [Γ2 : Γ2 ∩ gΓ1g−1] is finite then there is a natural

transformation:

[Γ2gΓ1] : H•(Γ1, ) −→ H•(Γ2, )

determined as being the unique such transformation compatible with the boundary homo-

morphisms and which coincide in degree zero with:

MΓ1 −→ MΓ2

m 7−→ ∑γigm

where:

Γ2 =∐

γi(Γ2 ∩ gΓ1g−1)

7

or equivalently:

Γ2gΓ1 =∐

γigΓ1

These are usually called Hecke operators, and the special case Γ1 ⊂ Γ2 and g = 1 is called

corestriction and denoted cor.

If Γ1, Γ2 are two groups and M1, M2 are modules over Γ1 and Γ2 respectively and, if

moreover, φ : Γ2 → Γ1 and θ :M1 →M2 satisfy:

θ((φγ)m) = γ(θm)

for all γ ∈ Γ2 and m ∈M1; then there is an induced map:

(φ∗, θ∗) : H•(Γ1,M1) −→ H•(Γ2,M2)

It may be defined as the unique map functorially extending:

θ :MΓ11 −→MΓ2

2

Returning to the situation in the last paragraph we may factor [Γ1gΓ2] as:

H•(Γ1,M)(c∗g−1 ,g∗)−→ H•(Γ2 ∩ gΓ1g

−1,M)cor−→ H•(Γ2,M)

where cg−1(γ) = g−1γg.

We shall now introduce a slight generalisation of Hecke operators. Let Γ1, Γ2 again be

subgroups of ∆ and g ∈ ∆ be such that [Γ2 : Γ2 ∩ gΓ1g−1] < ∞. Let M1, M2 be modules

for 〈g,Γ1〉 and Γ2 respectively. Let θ : gM1 →M2 be a map of Γ2 ∩ gΓ1g−1-modules. Then

we can define a map:

[Γ2gΓ1]θ : H•(Γ1,M1) −→ H•(Γ2,M2)

to be the composite:

H•(Γ1,M1)(c∗g−1 ,(θg)∗)−→ H•(Γ2,∩gΓ1g

−1,M2)cor−→ H•(Γ2,M2)

If we set M1 =M2 and θ = Id then we recover the normal Hecke operators.

8

Lemma 1.1 Let ∆ be a semi-group; Γ1 ⊃ Γ2 subgroups of ∆; g ∈ ∆ with [Γ1 : Γ1 ∩gΓ2g

−1] <∞; M1 (resp. M2) a module for 〈Γ1, g〉 (resp. 〈Γ2, g〉); and j :M1 →M2 a 〈Γ2, g〉morphism such that j : gM1

∼→ gM2. Then there exists I : H•(Γ2,M2)→ H•(Γ1,M1) such

that:

1. If there exist elements γi ∈ Γ1 such that Γ1gΓ1 =∐γigΓ1 and Γ1gΓ2 =

∐γigΓ2 then

I j∗ = [Γ1gΓ1].

2. If there exist elements δi ∈ Γ1 such that Γ1gΓ2 = (Γ2gΓ2)∐(∐

Γ2δigΓ2) and jδigM1 =

0 then j∗ I = [Γ2gΓ2].

The conditions in 1) and 2) are automatically satisfied if Γ1 = Γ2.

Proof:Set I = [Γ1gΓ2]j|−1gM1

. The first part is easy, it follows from the commutativity of

the following diagram:

H•(Γ1,M1)(c∗

g−1 ,g∗)

−→ H•(Γ1 ∩ gΓ1g−1, gM1) −→ H•(Γ1 ∩ gΓ1g

−1,M1)cor−→ H•(Γ1,M1)

j∗ ↓ j∗ ↓ ց res ց res ↑ cor

H•(Γ2,M2)(c∗

g−1 ,g∗)

−→ H•(Γ1 ∩ gΓ2g−1, gM2)

j∗∼= H•(Γ1 ∩ gΓ2g

−1, gM1) −→ H•(Γ1 ∩ gΓ1g−1,M1)

(The only slight problem is the right hand triangle, but working in the category of 〈Γ1, g〉-modules we need only check this in degree zero.)

For the second part we must check that the following diagram is commutative:

H•(Γ1,M1)j∗∼= H•(Γ1 ∩ gΓ2g

−1, gM1) −→ H•(Γ1 ∩ gΓ2g−1,M1)

cor−→ H•(Γ1,M1)

(c∗g−1 , g∗) ↑ j∗ ց ↓ j∗ ↓ j∗

H•(Γ2,M2)(c∗

g−1 ,g∗)

−→ H•(Γ2 ∩ gΓ2g−1,M2)

cor−→ H•(Γ2,M2)

The left hand side is easy, the only problem is to check that the two composite maps

H•(Γ3, N)→ H•(Γ2,M1) in:

H•(Γ3, N) −→ H•(Γ3,M1)cor−→ H•(Γ1,M1)

↓ j∗ ↓ j∗H•(Γ3 ∩ Γ2,M2)

cor−→ H•(Γ2,M2)

are equal, where Γ3 = Γ1 ∩ gΓ2g−1 and N = gM1. (If Γ1 = Γ2 this also is easy.) In fact we

shall prove this under the following assumptions, which are clearly valid in our case:

9

Γ3 ⊂ Γ1 and Γ2 ⊂ Γ1 are subgroups of ∆, M1 is a Γ1 module, N is a Γ3-submodule

of M1, M2 is a Γ2-module and j : M1 → M2 is a morphism over Γ2 such that Γ1 =

(Γ2Γ3)∐ (∐

Γ2δiΓ3) with jδiN = 0.

To prove this let φ ∈ Zn(Γ3, N) and let γ1, ..., γn ∈ Γ2. Let Γ2Γ3 =∐hk,0Γ3 and

Γ2δiΓ3 =∐hi,k,0Γ3 with each hk,0 and hi,k,0 in Γ2, so that jhi,k,0N = 0 and Γ2 =

∐hk,0(Γ2∩

Γ3). Then the image of φ in Hn(Γ2,M1) by the lower route is represented by:

(γ1, ..., γn) 7−→∑

hk,0j φ(h−1k,0γ1hk,1, ..., h−1k,n−1γnhk,n)

where hk,l is defined by h−1k,l−1γlhk,l ∈ Γ3 and hk,l = hk′,0 for some k′. Moreover if we define

hi,k,l in the similarly, then the image of φ by the upper route is represented by:

(γ1, ..., γn) 7−→ j∑hk,0φ(h

−1k,0γ1hk,1, ..., h

−1k,n−1γnhk,n)

+j∑hi,k,0φ(h

−1i,k,0γ1hi,k,1, ...)

=∑hk,0j φ(h−1k,0γ1hk,1, ..., h−1k,n−1γnhk,n)

1.2 Some Applications

The basic idea in the application of this lemma is that ifG is a reductive group, P a parabolic

subgroup, L a Levi component of P , A the split component of its centre, ΓP (N) the inverse

image of P (Z/NZ) under G(Z) → G(Z/NZ), M a G module with weights Φ ⊂ X∗(A),

φ0 ∈ Φ a lowest weight with respect to the partial order corresponding to P , and ν ∈ X∗(A)is such that ν.φ ≥ 0 for all φ ∈ Φ with equality if and only if φ = φ0; then one can define

a map j : M(Z/NZ) → Mφ0(Z/NZ) of ΓP (N)-modules such that j : ν(N)M(Z/NZ)∼−→

ν(N)Mφ0(Z/NZ) (here Mφ0 denotes the φ0 eigenmodule). By our lemma we then see that

H•(ΓP (N),M(Z/NZ)) depends up to the action of [ΓP (N)ν(N)ΓP (N)] only on the action

of L on Mφ0(Z/NZ). No doubt this can be formalised in this generality, but we shall simply

treat several examples.

Example 1.1

We shall prove:

10

Theorem 1.1 Fix a prime p and an extension of the p-adic valuation on Q to Qac (i.e.

Qac ⊂ Qacp ) and an integer N . Fix also a constant C. Then the sum of the dimensions of the

eigenspaces in Sk(Γ1(N)) for the Hecke operator Tp = [Γ1(N)

1 0

0 p

Γ1(N)] for which

the corresponding eigenvalue has p-adic valuation less than C is bounded independently of

k.

Note that we shall here employ the standard notation for elliptic modular forms. We

shall also use standard facts about them without comment. See for example [Sh2].

Proof:We first reduce to the case p|N . So suppose p 6 |N and without loss of general-

ity k > C2 + 1. If f ∈ Sk(Γ0(N), χ) is an eigenvalue for Tp with eigenvalue ap where

valp (ap) < C then the equation X2−apX+χ(p)pk−1 has a root α with valp (α) = valp (ap)

and a root β with valp (β) > C, and f(z) − βf(pz) ∈ Sk(Γ1(Np)) is an eigenvector for

[Γ1(Np)

1 0

0 p

Γ1(Np)] with eigenvalue α. Moreover if f1, ..., fr ∈ Sk(Γ1(N)) are lin-

early independent and if β1, ..., βr ∈ C then the functions fi(z) − βifi(pz) are linearly

independent in Sk(Γ1(Np)). The desired reduction now follows at once.

Thus assume p|N . By a theorem of Eichler and Shimura it will do to establish the theo-

rem withH1(Γ1(N), Sk−2((Qacp )2)) in place of Sk(Γ1(N)), and Tp = [Γ1(N)

p 0

0 1

Γ1(N)].

(Here Sn denotes the nth symmetric power.) In fact it will do to consider

H1(Γ1(Npr), Sk−2((Qac

p )2)) for any r ≥ 0 (because p|N implies that Tp commutes with

restriction from Γ1(N) to Γ1(Npr)). Let Bk(Qac

p ) denote the sum of the eigenspaces of

Tp in this cohomology group, which have p-adic valuation less than C. Also let Bk =

Bk(Qacp )∩H1(Γ1(Np

r), Sk−2(Z2p))

TF (where TF indicates the torsion free quotient). Then

Bk(Qacp ) = Bk ⊗Zp Q

acp . Take M = H1(Γ1(Np

r), Sn((Z/prZ)2)) for some fixed choice of

n > C and for r = n(n+ 1). Then for k ≥ n+ 2 we have a natural projection map:

j : Sk−2((Z/prZ)2) −→ Sn((Z/prZ)2)

11

as Γ1(Npr)-modules. If moreover g =

pn 0

0 1

we have:

j : gSk−2((Z/prZ)2)∼−→ gSn((Z/prZ)2)

and so by our lemma 1.1 we know that the kernel of:

Bk/prBk → H1(Γ1(Np

r), Sk−2((Z/prZ)2))j∗−→M

is killed by T np . Thus r(rkBk)− valp (detTnp ) ≤ valp (#M) so that r(rkBk) ≤ valp (#M) +

nC(rkBk) and rkBk ≤ valp (#M) as desired.

Before giving further examples we recall the notion of “Hida idempotent”. If M is a Zp

module with EndZp(M) a finite Zp-module (for example if M or Hom(M,Qp/Zp) is a finite

Zp-module) and if U :M →M is an endomorphism then there exists a unique idempotent

eU ∈ Zp[U ] ⊂ EndZp(M) such that:

1. U is invertible on eUM

2. U is topologically nilpotent on (1− eU )M

3. eU = limr→∞Ur!

4. if U commutes with another operator T so does eU

IfM ′ is a second such module with an operator U ′ and T :M →M ′ is such that TU = U ′T

then eU ′T = TeU . These results are all easy consequences of the discussion in [MW2]

(section 4). If A is a Zp-algebra we can think of eU ∈ EndA(M ⊗A). Moreover if M and U

are defined over Z, say M =M0 ⊗Z Zp then eU ∈ EndR(M0 ⊗Z R) where R = Zp ∩Qac. In

particular if we fix Qac ⊂ C and Qac ⊂ Qacp we may think of eU ∈ EndC(M0 ⊗ C). Exactly

similar results hold with Zp replaced by the completion of the integers of any number field

at any finite prime.

Note that we may deduce from the above example the following result of Hida:

Corollary 1.1 Fix a prime p, an integer N and embeddings Qac ⊂ C and Qac ⊂ Qacp . Let

e be the idempotent associated to the Hecke operator Tp (defined as above) on Sk(Γ1(N)).

Then dim eSk(Γ1(N)) is bounded independently of k.

12

We now consider our second:

Example 1.2

We recall some facts about Sp2g. Fix a maximal torus T in Sp2g(C) consisting of

diagonal matrices in the standard representation into GL2g. Fix X∗(T ) ∼= Zg by setting

~n = (n1, ..., ng) : (diag(µ1, .., µg, µ−11 , .., µ−1g )) = µn1

1 ...µngg . Also fix a Borel B consisting of

matrices of the form:

∗ 0 . . . 0 ∗ . . ∗∗ ∗ 0 . ....

. . .... . .

∗ ∗ . . . ∗ ∗ . . ∗0 . . 0 ∗ . . . ∗ ∗

. ....

. . ....

. . 0 ∗ ∗0 . . 0 0 . . . 0 ∗

Then the roots Φ of T on sp2g, the Lie algebra of Sp2g, are:

• αij the vector consisting of zeroes except for 1 in the ith place and −1 in the jth place

(i 6= j)

• βij the vector consisting of zeroes except for 1 in the ith and jth places (i > j)

• βii the vector consisting of zeroes except for 2 in the ith place

• γij = −βij (i ≥ j)

With respect to B the positive roots Φ+ are the αij with i > j and the βij . Note that:

αij =

ǫii − ǫjj 0

0 ǫjj − ǫii

βij =

ǫii + ǫjj 0

0 −ǫii − ǫjj

13

βii =

ǫii 0

0 −ǫii

γij = −βij

where ǫij is the g × g matrix with one in the ith row and jth column and zeroes elsewhere.

Also sp2g(C) has a Chevalley basis consisting of the following elements:

Xij =

ǫij 0

0 −ǫij

(i 6= j)

Yij =

0 ǫij + ǫji

0 0

(i > j)

Yii =

0 ǫii

0 0

Zij =t Yij (i ≥ j)

together with α(i+1)i for i = 1, ..., g − 1 and β11.

We shall let UZ denote the Z-subalgebra of the universal enveloping algebra of sp2g(C)

generated by elements of the formXnij

n! ,Y nijn! and

Znijn! . If V is an sp2g(C) module then by

an admissible lattice L ⊂ V we shall mean a Z lattice preserved by UZ. Then it is known

that any finite dimensional irreducible sp2g(C) module contains an admissible lattice, for

example UZv for any lowest weight vector v, and moreover that any admissible lattice is

equal to the sum of its intersections with the weight spaces of T in V .

Let R denote Z or Z/NZ. Let L be an admissible lattice in VL where ρL : Sp2g(C) →GLVL . We can define GL(R) to be the subgroup of GLL⊗R generated by elements of

the form expXij , exp Yij and expZij . We shall let PL(R) denote the subgroup of GL(R)

generated by the expXij and the expYij, and SL(R) the one generated by the expXij . It

is known that if L1 and L2 are as above with kerρL1 ⊂ kerρL2 then there is a unique map

GL1(R) → GL2(R) taking expW ∈ GL1(R) to expW ∈ GL2(R) for W equal to any Xij ,

Yij or Zij . (See [Sg].) We see that this map takes PL1(R) to PL2(R) and SL1(R) to SL2(R).

In particular we see that GL(R), PL(R) and SL(R) depend only on kerρL up to canonical

isomorphism.

14

If ρL is the standard (faithful) 2g dimensional representation of Sp2g(C) then GL(R) =

Sp2g(R), PL(R) = P (R) the subset of matrices of the form

A B

0 tA−1

with A ∈

SL2g(R) and SL(R) = SLg(R) the subset of these matrices with B = 0. We expect

that this is well known, but know no references. Briefly one shows by performing row and

column operations that Sp2g(R) is generated by matrices of the following forms:

1.

1− ǫij 0

0 1 + ǫij

= exp(−Xij) for i 6= j

2.

1g ǫij + ǫji

0 1g

= exp(Yij) for i > j

3.

1g ǫii

0 1g

= exp(Yii)

4.

1g 0

ǫij + ǫji 1g

= exp(Zij) for i > j

5.

1g − ǫii − ǫjj − ǫji + ǫij 0

0 1g − ǫii − ǫjj − ǫji + ǫij

= exp(Xij) exp(−Xji) exp(Xij) for i 6= j

6.

1g − ǫii ǫii

−ǫii 1g − ǫii

= exp(Yii) exp(−Zii) exp(Yii)

that P (R) is generated by matrices of types 1), 2), 3), and 5); and that SLg(R) is generated

by those of type 1) and 3).

Let:

Γ1(N) =

A B

C D

∈ Sp2g(R) | C ≡ 0 mod N detA ≡ 1 mod N

15

Then if L is any admissible lattice we have maps:

Sp2g(Z) −→ GL(Z/NZ)⋃ ⋃

Γ1(N) −→ PL(Z/NZ) −→ SL(Z/NZ)

compatible with the map L → L ⊗ Z/NZ. In particular Γ1(N) →→ SLg(Z/NZ) where

expXij 7→ expXij and expYij 7→ 0.

Recall that the irreducible representations of Sp2g(C) are parametrised by vectors ~n ∈X∗(T ) with 0 ≤ n1... ≤ ng. We shall denote the set of such vectors X∗(T )+. Let V~n denote

the Sp2g(C) module parametrised by ~n, and give it a GSp2g(C) action by letting µ12g act

by µ|~n|, where |~n| = ∑ni. Note that if ~a ∈ X∗(T ) is a weight of T on V~n then

µ1g 0

0 1g

acts on the corresponding weight space V ~a~n as µ12

∑(ni+ai) and that 1

2

∑(ni + ai) ∈ Z≥0. In

particular if µ ∈ Z then

µ1g 0

0 1g

preserves any admissible lattice.

Now choose v~n ∈ V~n a lowest weight vector. Set L~n = UZv~n, an admissible lattice in

V~n. Let V ′~n =⊕V ~a~n , where the sum is taken over weights ~a with

∑(ni + ai) = 0. Also let

U ′Z be the subalgebra of UZ generated by the elementsXnij

n! and L′~n = U ′Zv~n. Then L′~n ⊂ V ′~n.

In fact it is known that L~n is spanned over Z by vectors of the form∏i>j

Xaijij

aij !

∏i≥j

Ybijij

bij !v~n

and so we see that L′~n = L~n ∩ V ′~n and this is a direct summand of L~n (for an element of the

above form lies in V ′~n if and only if bij = 0 for all i, j).

Fix a positive integer N . L~n ⊗ (Z/NZ) is a Γ1(N)-module and this action factors

through PL~n . We can also make L′~n ⊗ (Z/NZ) a Γ1(N) module through the map Γ1(N)→SL~n(Z/NZ). I claim that with these actions the projection map:

j : L~n ⊗ (Z/NZ) −→ L′~n ⊗ (Z/NZ)

is a map of Γ1(N)-modules. But it will do to show that:

• j(Xijv) = Xijj(v)

• j(Yijv) = 0

16

and these are both clear. Moreover if g =

N1g 0

0 1g

then:

j : g(L~n ⊗ (Z/NZ))∼−→ g(L′~n ⊗ (Z/NZ))

Thus if Γ ⊂ Γ1(N) is of finite index, and if U denotes the Hecke operator [ΓgΓ] we see that

our proposition implies that there exists:

I : H•(Γ, L′~n ⊗ (Z/NZ)) −→ H•(Γ, L~n ⊗ (Z/NZ))

such that I j∗ = U and j∗ I = U . It is well known that these cohomology groups

are finitely generated abelian groups and so if N = p a prime we can associate a Hida

idempotent e to U . Then we have that:

j∗ : eH•(Γ, L~n ⊗ (Z/NZ))

∼−→ eH•(Γ, L′~n ⊗ (Z/NZ))

Now consider SLg ⊂ GSp2g by A 7→

A 0

0 tA−1

. Then we see from the fact that

V ′~n = L′~n ⊗ C that V ′~n is an irreducible slg(C)-module of heighest weight depending only

on the (g − 1)-tuple (n1 − n2, ..., n1 − ng). Thus if ~m ∈ Zg with 0 ≤ m1 ≤ ... ≤ mg and

mi−m1 = ni−n1 for i = 2, ..., g then we have an isomorphism of slg(C)-modules V ′~n∼→ V ′~m

such that v~n 7→ v~m, and so L′~n∼→ L′~m preserving the action of the Xij . Thus eH

•(Γ, L~n⊗Fp)depends up to canonical isomorphism only on the (g − 1)-tuple (n2 − n1, ..., ng − n1). We

deduce:

Theorem 1.2 Let p be a prime, Γ ⊂ Γ1(p) a subgroup of finite index. Fix Qac ⊂ Qacp and

Qac ⊂ C. Then we can associate a Hida idempotent e to the action of [Γ

p1g 0

0 1g

Γ]

on H•(Γ, V~n) for ~n ∈ X∗(T )+, and dim eH•(Γ, V~n+m~t) is bounded independently of m ≥ 0,

where ~t = (1, ..., 1) ∈ Zg.

Proof:Set ~m = ~n+m~t. Then it will do to show that dim eH•(Γ, L~m⊗Qp) is so bounded.

But we have seen that dim eH•(Γ, L~m ⊗ Fp) is so bounded and we have that:

17

• eH•(Γ, L~m ⊗ Zp)⊗ Fp → eH•(Γ, L~m ⊗ Fp)

• dim eH•(Γ, L~m ⊗ Zp)⊗Qp = dim eH•(Γ, L~m ⊗Qp)

so the result follows. (The first embedding comes from the long exact sequence correspond-

ing to 0→ Zpp→ Zp → Fp → 0.)

18

Chapter 2

Some Congruences between Siegel

Modular Forms

2.1 Introduction

In this chapter we are concerned with showing how, starting with a Siegel modular form f of

low weight which is an eigenform of the Hecke operators on a certain congruence subgroup

of Γ0(p) (those matrices in Sp2g(Z) congruent to

∗ ∗

0 ∗

mod p) which is ordinary in the

sense that it is an eigenvalue of a certain Hecke operator Up (see section 2.2) with eigenvalue

a p-adic unit; we can find a series of eigenforms of heigher weight whose eigenvalues under

the Hecke operators tend to those of the first form p-adically. We apply this to show how

standard conjectures about the existence of p-adic Galois representations corresponding to

such forms of genus two, if true for high weight (where one hopes to find the representations

in certain p-adic cohomology groups) would also be true for ordinary forms of low weight.

To explain our results more precisely recall that given g integers 0 ≤ n1 ≤ ... ≤ ng

we may consider Siegel modular forms of genus g and of weight ~n = (n1, ..., ng). These

are holomorphic functions on Siegel modular space valued in the irreducible representation

of GLg(C) with heighest weight ~n and which have certain transformation properties. If

12g(g + 1) ≤ n1 then such forms correspond to automorphic representations of GSp2g(A)

19

which are holomorphic discrete series at infinity. In this case one expects (conjecturally) to

be able to associate to certain such modular forms (those which are eigenforms of a Hecke

algebra) a system of 2g dimensional l-adic Galois representations, and to be able to find

these representations in the cohomology of certain sheaves on canonical models of certain

quotients of Siegel modular space. In the case n1 =12g(g+1)− 1 such forms correspond to

automorphic representations of GSp2g(A) which are limit of holomorphic discrete series at

infinity. One still expects to be able to associate Galois representations to such forms, but

one can no longer expects to be able to find them geometrically.

We shall show how to reduce the second case to the first in the case of “ordinary”

forms of genus two. The restriction on the genus is probably not important but it simplifies

some of the arguments and genus two is the case of most interest for us. An eigenform on a

congruence subgroup Γ of Γ0(p) is called ordinary at p if its eigenvalue for the Hecke operator

Up = [Γ

1g 0

0 p1g

Γ] is a p-adic unit. An eigenform of genus two for a congruence

subgroup Γ of Sp4(Z) which is dense in Sp4(Zp) is called ordinary at p if a certain quartic

polynomial Qp(X) (see 2.4) associated to f has distinct roots the ratio of no two of which

equals p and one of which is a p-adic unit (in the case n1 ≥ 2 this is certainly true if Qp(X)

has two distinct roots which are p-adic units).

Our method is as follows. One can write down a theta series θ of weight (p−1, ..., p−1)

which is congruent to one modulo p. Multiplying a form f by high p powers of θ produces

very congruent forms of heigher weight. This along with some background on Siegel modular

forms is discussed in section 2.2. Unfortunately this does not seem to be enough for the

applications to Galois representations, the problem being that if one starts with an eigenform

of the Hecke operators one does not obtain an eigenform highly congruent to it. To overcome

this problem in the case of a congruence subgroup contained in Γ0(p) one uses the fact that

the number of ordinary eigenforms on Γ of level ~n+m(1, ..., 1) is bounded independently of

m. This is proved by embedding the cusp forms in a certain cohomology group and using

the results of the first chapter to relate these as m varies. From this it is not difficult to

see (by an argument of Wiles using Fitting ideals) that we can lift fθpm

to an eigenform

20

of weight ~n+ pm(p− 1)(1, ..., 1) with eigenvalues congruent to those of f modulo pm+1C for

some C independent of m, as we wanted. This argument is discussed in section 2.3. The

application to Galois representations is discussed in section 2.5. In section 2.4 we show

how to construct from an eigenform Γ on a general congruence subgroup which is dense

in Sp4(Zp) an eigenform on Γ ∩ Γ0(p) which is also an eigenform for Up. We use this to

generalise the results about Galois representations to congruence subgroups not contained

in Γ0(p).

2.2 Review of Siegel Modular Forms

Fix an integer g ≥ 1. Let GSp2g(R) denote the set of α ∈ GL2g(R) such that:

α

0 1g

−1g 0

tα = ν(α)

0 1g

−1g 0

for some ν : GSp2g(R)→ R×. Let Sp2g(R) be those elements of GSp2g(R) in the kernel of

ν. Let G(R) = GSp2g(R), G∞ = G(R), G+∞ = ν−1R×>0 ⊂ G∞, G(Q)+ = G(Q) ∩ G+

∞, and

A (resp. Af ) denote the adeles (resp. finite adeles) of Q. Let U∞ denote the group of: A B

−B A

∈ GL2g(R)

such that AtB is symmetric and AtA+BtB is a nonzero scalar, or:

U∞ =(α, ν) ∈ GLg(C)×C× | (α, ν)∗ = (α, ν)

where (α, ν)∗ = (νc(tαc)−1, νc) (c denoting complex conjugation), and the correspondence

is given by:

β =

A B

−B A

7−→ (A− iB, ν(β))

Thus U∞ is a real form of GLg(C)× C×. It is in fact the g × g unitary similitudes.

Let Z = Zg denote the set of symmetric complex g×g matrices x+√−1y with y positive

definite. Then G+∞ acts on Z by

A B

C D

: z → (Az +B)(Cz +D)−1. If z0 = (

√−1)1g

21

then α 7→ αz0 gives a bijection G+∞/U∞

∼→ Z. We define a map J : G+∞×Z → GLg(C)×C×

by: α =

A B

C D

, z

−→ (Cz +D, ν(α))

so that:

J(αβ, z) = J(α, βz)J(β, z)

If ρ is a finite dimensional representation of GLg(C) × C× on a complex vector space V

then we set Jρ = ρ J : G+∞ ×Z → Aut(V ).

If U ⊂ G(Af ) is an open compact subgroup we let Sρ(U) denote the space of functions

φ : G(Q)\G(A)→ V such that:

• φ(guu∞) = ρ(u∞)−1φ(g) for all g ∈ G(A), u ∈ U and u∞ ∈ U∞

• if h ∈ G(A) then the function:

fh : Z −→ V

αz0 7−→ Jρ(α, z0)φ(hα)

where α ∈ G+∞, is holomorphic. (It is easily checked that this function is well defined.)

•∫N(Q)\N(A) φ(nh)dn = 0 where N is the unipotent radical of any proper parabolic

subgroup and where dn is any invariant measure on N(Q)\N(A)

We set Sρ =⋃Sρ(U) as U ranges over open compact subgroups. Then G(Af ) acts on Sρ on

the right by (φ|g)(h) = φ(hg−1), and Sρ(U) = SUρ . We define similarlyMρ(U) by omitting

the last condition (assuming g > 1, which is the only case we shall be concerned with as

the case g = 1 is well known).

Similarly if Γ ⊂ G+∞ is a discrete subgroup we set Sρ(Γ) to be the set of holomorphic

functions f : Z → V such that:

• f |γ = f for all γ ∈ Γ

• limλ→+∞(f |γ)

z 0

0 iλ

= 0 for all γ ∈ G(Q)+ and all z ∈ Zg−1

22

where for γ ∈ G+∞ we define:

(f |γ)(z) = Jρ(γ, z)−1f(γz)

Similarly we may define Mρ(Γ) by dropping the last condition (if g > 1).

Assume U ⊂ G(Af ) is an open compact subgroup such that (νU)Q×R×>0 = A×. Then

if we set ΓU = U ∩G(Q)+ we have an isomorphism:

Sρ(U) ∼= Sρ(ΓU )

given by:

φ 7−→ (fφ : αz0 7→ Jρ(α, z0)φ(α))

and inversely by:

f 7−→ (φf : γuα 7→ Jρ(α, z0)−1f(αz0))

where α ∈ G+∞, u ∈ U and γ ∈ G(Q). The second map is well defined as, by the strong

approximation theorem and our assumption on U , we have that G(A) = G(Q)UG+∞. If

h ∈ G(Af ) and h = uγ with u ∈ U and γ ∈ G(Q)+ then:

φf |h = φh|γ and fφ|h = fφ|γ

Now let U and U ′ be open compact subgroups of G(Af ) and let g ∈ G(Af ), then we

define a Hecke operator:

[UgU ′] : Sρ(U) −→ Sρ(U ′)φ 7−→ ∑

φ|giwhere UgU ′ =

∐Ugi. If U and U ′ also satisfy the condition of the last paragraph we can

think of [UgU ′] : Sρ(ΓU ) → Sρ(ΓU ′). It is given by f 7→∑f |γi where gi = uiγi with

γi ∈ G(Q)+ and ui ∈ U . Equivalently we may write g = uγ with γ ∈ G(Q)+ and u ∈ U ,

and then the γi’s may be defined by ΓUγΓU ′ =∐

ΓUγi.

It is well known that if V is irreducible there is an inner product on V , say 〈 , 〉, suchthat:

〈ρ(α)v1, ρ(α)v2〉 = ν(α)µ〈v1, v2〉

23

for all α ∈ U∞ and v1, v2 ∈ V , and where µ depends only on ρ (in fact ρ(x1g, x2) = xµ).

Now define an inner product on Sρ(U) by:

〈φ1, φ2〉 =∫

G(Q)\G(A)/UU∞

〈φ1(g), φ2(g)〉||ν(g)||−µdg

where dg is an invariant measure on G(Q)\G(A)/UU∞ and ||.|| : Q×\A× → R×>0 by

x 7→ ∏ |xv|v. This is easily checked to be well defined. Moreover the adjoint of g is

||ν(g)||−µg−1 (if the two measures are normalised correctly) and the adjoint of [UgU ′] is

||ν(G)||−µ[U ′g−1U ].

We now introduce some specific representations ρ. We recall first some multilinear

algebra. Let 0 ≤ n1 ≤ ... ≤ ng be integers, and set ~n = (n1, ..., ng) and |~n| = n1 + ...+ ng.

Also let ~t = (1, ..., 1). S|~n|, the symmetric group on |~n| letters acts on M⊗|~n|. There is an

element c in Z[S|~n|] (unique up to ±1) satisfying:

• c2 = µc for some scalar µ

• c = ∑σ∈S|~n|

δσσ where δσ = 0, 1 or −1

• cσ = c if σ preserves the sets 1, ..., n1, n1 + 1, ..., n1 + n2, etc.

• σc = (−)σc (where (−)σ denotes the sign of sigma) if σ preserves sets of the form

nij−1 + j, nij−1+1 + j, ..., |~n| − ng + j where ij is the least index such that nij ≥ j

(where n0 = 0).

We can think of c ∈ End(M⊗|~n|). Let⊗~nM denote cM⊗|~n|. Then

⊗~n commutes with

localisation and if α : M → N is linear we get a linear map ⊗~n(α) : ⊗~n(M) → ⊗~n(N).

If R is any ring this gives a natural action of GLg(R) on⊗~n(Rg), and if R is a field of

characteristic zero this representation is known to be irreducible (see [We]).

Let W~n denote the GLg(R)×R× module⊗~n(Rg) with the above action of GLg(R) and

on which R× acts via λ 7→ λ12g(g+1)−|~n|. Let ρ~n denote the corresponding representation,

let W~n denote W~n(C) and drop the ρ when ρ~n is used as a sub- or super-script.

If R is a ring denote the ring of formal power series:

∑

h∈symm∗g(Z)≥0

ahqh

24

in an indeterminate q by R[[q]]g. Here symm∗g(Z) denotes the semigroup of g×g symmetric

integral matrices with even diagonal entries, and a superscript ≥ 0 (resp. > 0) indicates

those which are positive semi-definite (resp. positive definite). If M is an R-module we

define M [[q]]g = R[[q]]g ⊗M .

If f ∈Mρ(Γ) there is an integer N depending only on Γ such that f(z+Nh) = f(z) for

h ∈ symmg(Z). Thus we have a Fourier expansion:

f(z) =∑

h∈symm∗g(Z)

≥0

ah(f) exp(π√−1N−1tr(hz))

and so we get an embedding Mρ(Γ) → W~n[[q1/N ]]g. f ∈ Sρ(Γ) if and only if ah(f |γ) = 0

for all γ ∈ G(Q)+ and for all det h = 0. If R ⊂ C we define M~n(Γ, R) = Mρ~n(Γ) ∩W~n(R)[[q

1/N ]]g. We define S~n(Γ, R) similarly, and for U an open compact subgroup ofG(Af )

with (νU)Q×R×>0 = A× we define S~n(U,R) to be S~n(ΓU , R), and similarly forM~n(U,R).

Lemma 2.1 Let Γ ⊂ G(Q)+ be a discrete congruence subgroup, then there is a finite abelian

extension K/Q such that S~n(Γ) = S~n(Γ,OK)⊗C, and similarly M~n(Γ) =M~n(Γ,OK)⊗C.

Proof:It will clearly do to show the following:

1. M~n(Γ) is finite dimensional

2. M~n(Γ) =M~n(Γ,Qab)⊗ C

3. S~n(Γ) = S~n(Γ,Qab)⊗ C

4. if f ∈M~n(Γ,Qab) then there exists 0 6= C ∈ Qab with Cf ∈M~n(Γ,OQab)

1) is well known. If M~n denotes the union over all congruence subgroups Γ of M~n(Γ) then

Shimura has proved that M~n =M~n(Q)⊗C and that G(Q)+ preserves M~n(Qab) (see [Sh3]).

2) and 3) follow from this. Finally it will do to establish 4) in the special case when Γ

is equal to the set of matrices in Sp2g(Z) which are congruent to 12g modulo N for some

N > 3.

It is known (see for example [Fa2]) that there is a separated schemeM and a principally

polarised abelian scheme A/M of relative dimension g together with an isomorphism α :

25

(µgN × (Z/NZ)g) ×ZM ∼→ A[N ] taking the standard pairing (µgN × (Z/NZ)g)2 → µN to

the Weil pairing A[N ]2 → µN (i.e. a level N structure), with the property that if A/S is

a principally polarised abelian scheme with level N structure, then there is a unique map

S →M such that A ∼= A×M S and the polarisation and level N structure on A come by

pulling back that on A. If A/S is an abelian scheme, let ωA/S denote the direct image of

the sheaf of relative differentials ΩA/S . Then ωA/S is quasicoherent on S, and if T → S

then ωA×T/T is the inverse image of ωA/S. (These are easy from the definition of Ω and the

fact that A→ S is quasicompact.) In particular ωA/S is the inverse image of ωA/M under

the canonical map S →M, and we get a map ωA/M(M)→ ωA/S(S).

Now consider Z[[q1/N ]]. There is an h ∈ symm∗g(Z) such that if f = qh and R =

Z[[q1/N ]][f−1], then it is known (the Mumford construction, but see [Fa2]) that there is an

abelian scheme A/spec R with a canonical isomorphism ωA/R ∼= Rg. Thus we obtain a map

ωA/M(M) → Rg. Moreover if ~n is as above we get a map (⊗~n ωA/M)(M) →⊗~n(Rg). If

we tensor over Z with C we get a commutative diagram:

(⊗~n ωA/M)(M) −→ ⊗~n(Rg)

↓ ↓(⊗~n ωAC/MC

)(MC) −→⊗~n(R⊗ C)g

It is further known that (⊗~n ωAC/MC

)(MC) =M~n(Γ) and that the map (⊗~n ωAC/MC

)(MC)

→ ⊗~n(C[[q1/N ]]g[f−1]g) is just the normal q-expansion. Finally as ωA/M is quasicoher-

ent and C is flat over Z,⊗~n(ωAC/MC

(MC)) =⊗~n(ωA/M(M)) ⊗ C. Thus the image of

⊗~n(ωA/M(M)) in M~n(Γ) spans M~n(Γ) and each element has a Fourier expansion with

coefficients in Z as desired.

Corollary 2.1 With the notation as in the lemma, if Γ = ΓU for U ⊂∏GSp2g(Zl) an

open compact subgroup normalised by i∏

Z×l where i : Gm → GSp2g by t 7→

1g 0

0 t1g

,

then we may take K = Q.

Proof:It will do to show that S~n(Γ,Qab) is stable under the action of Gal(Qab/Q), and

similarly for M~n(Γ,Qab). We shall only treat the first case, which is marginally harder.

26

Shimura (see [Sh3]) defines an action of G(Af ) on M~n(Qab) such that in particular every

function is stabilised by an open subgroup, G(Q)+ has its normal action, and if t ∈ ∏Z×l

then f it = f (t−1,Qab/Q), where ( ,Qab/Q) is the Artin symbol. Let f ∈ S~n(Γ,Qab) and

t ∈∏Z×l , we must show that f it ∈ S~n(Γ,Qab).

Firstly let α ∈ G(Q)+ and h ∈ symm∗g(Z)≥0 with deth = 0. Then we can find u ∈

stabG(Af )(f), β ∈ G(Q)+ and s ∈∏Z×l such that i(t)α = uβi(s), so we see that:

ah(fi(t)|α) = ah(f

uβi(s)) = ah(f |β)(s−1,Qab/Q) = 0

Thus f i(t) ∈ S~n(Qab). Secondly let α ∈ Γ, then we can find β ∈ G(Q)+, s ∈ ∏

Z×l and

u ∈ W = x ∈∏G(Zl)|x ≡ 12g mod N ⊂ stabG(Af )(f) ∩ U for some N , such that

i(t)α = uβi(s). Then we see that β ∈ Sp2g(Z).1, i(−1) and ν(u)s = ±t. Without loss of

generality we may assume s = t, ν(u) = 1 and β ∈ Sp2g(Z). In fact in this case:

β ∈ Sp2g(Z) ∩W.i(t)Ui(t)−1

⊂ Sp2g(Z) ∩ U = Γ

and so:

f i(t)|α = fuβi(t) = (f |β)i(t) = f i(t)

and we are done.

We shall now introduce some particular Hecke operators of special importance for us. Fix

a rational prime p and consider an open compact subgroup U = U1×U2 ⊂∏l 6=pGSp2g(Zl)×

GSp2g(Zp) satisfying:

• U ⊃

1g 0

0 (∏

Z×l )1g

• there is an integer r = r(U) ≥ 1 such that U2 is the set of matrices

A B

C D

∈

GSp2g(Z) with C ≡ 0 mod pr and (A mod pr) and (D mod pr) lying in some finite set

of possibilities.

Let N ′ = N ′(U) be the smallest positive integer such that U contains all elements of∏GSp2g(Zl) which are congruent to one modulo N ′, and write N ′(U) = N(U)pr (or if no

27

confusion can arrise N ′ = Npr. Then we define:

Up = [U

1g 0

0 p1g

U ] = [U

1g 0

0 πp1g

U ]

where πp denotes a uniformizer in Zp. Then we have:

Lemma 2.2 Let U be as described above. Then S~n(U) = S~n(U,Z) ⊗ C and the Hecke

operator Up preserves S~n(U,Z). In fact (∑ahq

h/N )|Up =∑aphq

h/N . Thus we can define

a Hida idempotent e on S~n(U) as in section 1.2.

Proof:Let X be a set of representatives for symmg(Z) modulo p, such that each X ∈ Xis congruent to zero modulo N . Then:

U

1g 0

0 p1g

U =

∐

X∈X

U

1g X

0 p1g

as follows easily from the fact that if

A B

C D

∈ U there is an X ∈ X with B ≡

AX mod p (A is invertible modulo p) and from the equality: 1g 0

0 p1g

A B

C D

=

A p−1(B −AX)

pC D −CX

1g X

0 p1g

Now note that for h ∈ symm∗g(Z):

∑

X∈X

exp(π√−1N−1p−1tr(hX)) =

p12g(g+1) if h ∈ p symm∗g(Z)

0 otherwise

(If h ∈ p symm∗g(Z) this is clear. If not pick Y ∈ X with 2Np 6 |tr(hY ), and then:

∑X∈X exp(π

√−1N−1p−1tr(hX)) = (

∑X∈X exp(π

√−1N−1p−1tr(hX)))

exp(π√−1N−1p−1tr(hY ))

and the result follows.) Now:

(∑ah exp(π

√−1N−1tr(hz)))|Up

= ρ~n(p1g, p)−1

∑ah exp(π

√−1N−1p−1tr(hz))∑X exp(π

√−1N−1p−1tr(hX))

= p−12g(g+1)∑

p|h ahp12g(g+1) exp(π

√−1N−1p−1tr(hz))

=∑aph exp(π

√−1N−1tr(hz))

28

Keep the notation of the lemma. Then Z = Z(U) = ((Z/N ′Z)×)g acts on S~n(U) by

(a1, ..., ag) 7→ σa = diag(a1, ..., ag , a1−1, ..., ag

−1) where ai ∈∏

Z∗l with (ai)l equal to 1 if

l 6 |N ′ and ai if l|N ′. We can decompose S~n(U) =⊕

χ∈Z S~n(U)χ. Let R = Qab ∩ OQacp , and

S~n(U,χ,R) = S~n(U)χ ∩ S~n(U,R). It follows from the results of Shimura discussed above

that S~n(U)χ = S~n(U,χ,U) ⊗R C. Let T = T(U) be the abstract double coset algebra over

Z generated by the operators [UxU ] where x ∈ M2g(Zl) ∩ GSp2g(Ql) for all primes l 6 |N ′.It is known that T is commutative (see for example [A2]). Moreover T acts on S~n(U), and

each element T ∈ T acts as a normal operator (i.e. it commutes with its adjoint). Thus

the elements of T can be simultaneously diagonalised. The action of T commutes with that

of Z, and if r(U) ≥ 1 these both commute with the action of Up. I claim that T preserves

S~n(U,χ,R).To see this let x ∈ M2g(Zl) ∩ GSp2g(Q) then we can write UxU =

∐Uxi where

xi ∈M2g(Zl) ∩GSp2g(Ql) is of the form:

a1 0 . . . 0 ∗ ∗ . . . ∗∗ a2 0 ∗ ∗ ∗...

. . ....

.... . .

...

∗ ∗ . . . ag ∗ ∗ . . . ∗0 0 . . . 0 ba−11 ∗ . . . ∗0 0 . . . 0 0 ba−12 . . . ∗...

. . ....

.... . .

...

0 0 . . . 0 0 0 . . . ba−1g

(see for instance [A2]). We may further suppose that b and each ai lies in lZ ⊂ Zl. Then

consider an element x′i ∈M2g(Z) ∩GSp2g(Q) defined by:

• x′i lies in the same Borel as xi was required to lie in

• x′i ≡ diag(a1, ..., ag , ba−11 , ..., ba−1g ) mod N ′

• x′i ≡ xi mod l?

29

Then for ? large enough Uxi = Uσa−1x′i so that f |xi = χ(a−1)f |x′i and x′i clearly preserves

R[[q1/N ]]g.

Finally we introduce a particular modular form which we shall need later.

Lemma 2.3 Fix a rational prime p, and set:

Γ0(p) =

A B

C D

∈ Sp2g(Z)

∣∣∣∣∣∣C ≡ 0 mod p

Then there is an element θ ∈M(p−1)~t(Γ0(p),Z) with a0(θ) = 1 and p|ah(θ) for all h 6= 0.

Proof:This argument is due to Hida in the case g = 1.

Let Qn denote the (n− 1)× (n− 1) matrix:

2 −1 0 . . 0

−1 2 −1 0

0 −1 2 0

. . .

. . .

0 0 0 . . 2

then Qn ∈ symm∗n−1(Z) and an easy induction on n shows that detQn = n and hence in

particular Qn is positive definite. Let:

θ1 =∑

X∈M(p−1),g(Z)

exp(π√−1tr(tXQpXz))

Then theorems 2 and 3 of [AM] imply that θ = θ21 ∈ M(p−1)~t(Γ0(p),Z). It is clear that

a0(θ) = a0(θ1)2 = 1. Now let ζ be the (p− 1)× (p − 1) matrix:

0 0 0 . . 0 −11 0 0 0 −10 1 0 0 −1. . .

. . .

0 0 0 0 −10 0 0 . . 1 −1

30

It has characteristic polynomial Xp−1 + ... + 1 and its eigenvalues are the nontrivial pth

roots of one. Thus Z/pZ acts on M(p−1),g(Z) by m : X 7→ ζmX and the orbits are either

0 or have cardinality p. It is easily checked that tζQpζ = Qp, so that tXQpX is constant

on such orbits. Then p|ah(θ1) for h 6= 0, from which the result follows.

Corollary 2.2 Let U be an open compact subgroup of G(Af ) satisfying the conditions de-

scribed before lemma 2.2. Let R = Qab ∩ OQacp and χ be a character on Z(U). Then there

is a map:

im : eS~n(U,χ,R) → eS~n+(p−1)pm−1~t(U,χ,R)

such that for all h ∈ symm∗g(Z) ah(imf) ≡ ah(f) mod pm and infact for all T ∈ T(U) we

have ah((imf)|T ) ≡ ah(f |T ) mod pm. (Recall that ~t = (1, ..., 1).)

Proof:Set im(f) = e(θpmf). Then θp

mf ≡ f mod pm, so U r!p (θ

pmf) ≡ U r!p f mod pm and

hence e(θpmf) ≡ ef = f mod pm.

Also if T ∈ T recall that f |T =∑ζif |x′i where ζi is a root of unity and x′i ∈M2g(Z) ∩

GSp2g(Q) is as described in the discussion after lemma 2.2. But if |~n denotes the action as

for modular forms of weight ~n, then:

(∑

ahqh/N )|~n+a~tx′i = λai (

∑ahq

h/N )|~nx′i

for some integer λi depending only on x′i. Thus:

im(f)|~n+(p−1)pm−1~txi ≡ im(f |~nxi) mod pm

and so im(f)|T ≡ im(f |T ) mod pm as desired.

2.3 Relation to Cohomology

Our aim in this section is to relate our spaces of automorphic forms to certain cohomol-

ogy groups and use this to prove an analogue of theorem 1.2 for automorphic forms. We

then apply this to find congruences between eigenforms of the Hecke operators in different

weights.

31

We fix some notation:

sp2g =

A B

C −tA

∈M2g

∣∣∣∣∣∣tB = B, tC = C

k =

A B

−B A

∈M2g

∣∣∣∣∣∣tB = B, tA = −A

p =

A B

B −A

∈M2g

∣∣∣∣∣∣tB = B, tA = A

a =

0 Λ

−Λ 0

∈M2g

∣∣∣∣∣∣Λ is diagonal

Then sp2g = k ⊕ p, k(R) is the Lie algebra of K∞ = U∞ ∩ Sp2g(R), and a ⊂ k ⊂ sp2g is a

Cartan subalgebra. We fix a(C)′ ∼= Cg by ~x : diag(λ1, ..., λg) 7→√−1

∑xiλi. Then if T

is the maximal torus of Sp2g(C) considered in example 1.2, and t(C) is its Lie algebra, we

may conjugate t(C) to a(C) such that the map:

Zg ∼= X∗(T ) ⊂ t(C)′∼−→ a(C)′ ∼= Cg

is the canonical inclusion. Thus the roots of a on sp2g are just the vectors αij (i 6= j), βij

(i ≤ j) and γij (i ≤ j) as in example 1.2. The roots Φc of a on k are the αij , and those

(Φn) on p are the βij and the γij. Choose the same order we chose in example 1.2. Let

W denote the Weyl group of sp2g and Wn the subset of elements w such that Φ+c ⊂ wΦ+

where Φ+c = Φc ∩ Φ+. Let U denote the universal enveloping algebra of sp2g(C), and Z(U)

its centre. Then the homomorphisms Z(U)→ C are parametrised by a(C)′/W (the Harish-

Chandra parametrisation). We shall denote this correspondence λ ↔ χλ. We now restate

a special case of theorem 10 of [Fa1] in a slightly different notation:

Let:

• Γ ⊂ G+∞ be a torsion free discrete subgroup

• w ∈Wn of length l(w)

• ∆ be the basis corresponding to Φ+, and δG = (1, ..., g) half the sum of the

positive roots

32

• λ ∈ a(C)′ be a dominant (integral) weight with respect to ∆

• W (µ) denote the irreducible K∞ module with highest weight µ

• Cµ be the space of cuspidal C∞ functions f : Γ\Sp2g(R)→W (µ) such that

f(hk) = k−1f(h) for all h ∈ Sp2g(R) and k ∈ K∞

• Cχνµ be the subspace of Cµ that transform by χν under the action of Z(U)

• V (λ) be the irreducible Sp2g(C) module of heighest weight λ

• V(λ) the sheaf on Γ\Z defined by setting V(λ)(U) to be the set of C∞

functions f : U → V (λ) such that f(γz) = γf(z) for all z ∈ Z and γ ∈ Γ,

and where U is the pre-image of U under the map Z → Γ\Z

• H•P denote the image of the cohomology of compact support in the coho-

mology, or equivalently the kernel of the map from the cohomology to the

cohomology of the boundary of the Borel-Serre compactification

then:

Cχλ+δGw(λ+δG−w−1δG)

→ Hl(w)P (Γ\Z,V(λ)) → H l(w)(Γ, V (λ))

Moreover the Hecke operator [ΓgΓ] corresponds to the Heck operator [Γg−1Γ],

for g ∈ Sp2g(R) for which these operators make sense.

Now take w : (x1, ..., xn) 7→ (−xn, ...,−x1). Note that δG − w−1δG = (g + 1)~t. Let

δK = 12(1−g, 3−g, ..., g−1) be half the sum of the elements of Φ+

c . Then the representation

ρ~n of K∞ ⊂ U∞ defined in the last section has heighest weight w~n, which is dominant

with respect to wΦ+. Thus Z(U) acts on S~n(Γ) via χw~n+2wδK−wδG = χw~n+δG so that for

Γ torsion free and for ~n with g + 1 ≤ n1 ≤ ... ≤ ng:

S~n(Γ) → Cχ~n+w−1δGw~n → H

12g(g+1)

P (Γ\Z,V(~n − (g + 1)~t))

→ H12g(g+1)(Γ, V (~n− (g + 1)~t))

and so for any discrete Γ:

S~n(Γ) → H12g(g+1)(Γ, V (~n− (g + 1)~t))

33

(We may choose a normal subgroup Γ′ of finite index which is torsion free and then take Γ/Γ′

invariants, using the inflation restriction sequence on the right.) Moreover if g ∈ G(Q)+ then

the action of [ΓgΓ] on S~n(Γ) corresponds to that of [ΓıgΓ] on H12g(g+1)(Γ, V (~n− (g+1)~t)),

where ıg =

0 −1g

1g 0

tg

0 −1g

1g 0

−1

. To see this we need only check that for

µ ∈ R×>0 [Γµ12gΓ] acts on both by µ|~n|−g(g+1), which is easy. Now we have:

Proposition 2.1 Let U ⊂ G(Af ) be an open compact subgroup satisfying the conditions

stated before lemma 2.2 with r(U) ≥ 1, let ~n be such that 0 ≤ n1... ≤ ng and let e denote

the Hida idempotent. Then there is a constant C such that:

dim eS~n+m~t(U) < C

for all m ≥ 0, where ~t = (1, ..., 1).

Proof:Without loss of generality we may restrict to m ≥ g + 1. Then by theorem 1.2

we can choose C such that dim eH12g(g+1)(ΓU , V (~n+m′~t)) < C for all m′ ≥ 0. Now fix m.

Also choose a finite Z module M ⊂ C such that:

S~n+m~t(ΓU ,Z) → H12g(g+1)(ΓU , L~n+(m−g−1)~t)

TF ⊗M

Then:

eS~n+m~t(Γ,Z) → e(H12g(g+1)(ΓU , L~n+(m−g−1)~t)

TF ⊗M)

= (eH12g(g+1)(ΓU , L~n+(m−g−1)~t)

TF )⊗M⊂ eH

12g(g+1)(ΓU , V~n+(m−g−1)~t)

and so dimS~n+m~t(ΓU ,Z) < C and we are done.

We can now deduce our first main result:

Theorem 2.1 Let U ⊂ G(Af ) be an open compact subgroup satisfying the conditions stated

before lemma 2.2 with r(U) ≥ 1, let ~n be such that 0 ≤ n1 ≤ ... ≤ ng, let R = Qab∩OQac, and

let e denote the Hida idempotent. Let f ∈ eS~n(U)χ be an eigenform for T with eigenvalues

given by λ : T→ R. Then we can find fm ∈ eS~n+am~t(U)χ such that:

34

• fm is an eigenvalue for T with eigenvalues λm : T→ R

• am →∞ as m→∞

• supT |λ(T )− λm(T )|p → 0 as m→∞

Proof:We may assume f ∈ eS~n(U,χ,R) and that if µ ∈ Qab and µf ∈ S~n(U,χ,R) thenµ ∈ R. Then by corollary 2.2 we can find f ′m ∈ eS~n+am~t(U,χ,R) with fm|T ≡ f |T mod pr

′m

for all T ∈ T, where am and r′m tend to infinity with m. Let Tm denote the image of T in

End(eS~n+am~t(U,χ,R)). Then we get λ′m : Tm →→ R/pr′mR with λ′m ≡ λ mod prm. Call its

kernel Im. Let C be the bound from the last proposition (proposition 2.1). Then we can find

less than C functions hm,i ∈ eS~n+am~t(U,χ,R) which span eS~n+am~t(U,χ,R) and such that

each hm,i is an eigenform for Tm with eigenvalues given by λm,i say. Then Vm = ⊕Rhm,i is afaithful Tm module. Thus, if Fitt denotes the Fitting ideal (see, for example, the appendix

of [MW1]), FittTm(Vm) = 0 and so:

0 = FittTm/Im(Vm/ImVm)

=∏i FittR/pr′mR(R/λm,i(Im))

=∏i λmi(Im) ⊂ R/pr

′mR

Thus for some i, valp(λm,iIm) ≥ r′m/C. Let fm = hm,i and we are done.

2.4 Some Lemmas on Hecke Operators

The discussion in the other sections is principally concerned with “ordinary” forms, i.e.

modular forms in the image of the Hida idempotent e acting on a space of modular forms for

an open compact subgroup contained in U0(p), the set of matrices

A B

C D

in

∏G(Zl)

with C ≡ 0 mod p. It is often more interesting to consider a modular form f on an open

compact subgroup containing G(Zp). In this section we give a criterion for ef ∈ S(U∩U0(p))

not to vanish. In fact we shall only treat the case of genus two. The calculations are already

very messy, and this is the case of principal interest for us.

35

In the case of genus one the answer to this question is very easy. If f ∈ Sk(Γ1(N)),

with N prime to p, is an eigenform for Tp and Sp, say with eigenvalues ap and dp, then

ef ∈ Sk(Γ1(Np)) is non-zero if and only if one root of the equation Q(X) = X2−apX+pdp

is a p-adic unit. To prove this (at least in the case that this polynomial has distinct roots)

one writes down two forms f1 and f2 in Sk(Γ1(Np)) which are both eigenforms for Up with

eigenvalues the roots of Q(X), and such that f is a linear combination of the two (and not

a multiple of either one separately).

For the rest of this chapter we assume that g = 2. Also let N be an integer and U the

open compact subgroup of∏GSp4(Zl) consisting of matrices congruent to

12 0

0 λ12

modulo N . Let p be a prime not dividing N . Then we define Hecke operators:

• T (p) = Tp =

U

1 0 0 0

0 1 0 0

0 0 πp 0

0 0 0 πp

U

• Tp2 =

U

1 0 0 0

0 1 0 0

0 0 π2p 0

0 0 0 π2p

U

• Rp =

U

1 0 0 0

0 πp 0 0

0 0 π2p 0

0 0 0 πp

U

• Sp =

U

πp 0 0 0

0 πp 0 0

0 0 πp 0

0 0 0 πp

U

• T (p2) = Tp2 +Rp + Sp

36

where πp denotes a uniformiser in Zp. Also let Qp(X) be the formal polynomial whose

coefficients are Hecke operators given by:

X4 − TpX3 + (T 2p − T (p2)− p2Sp)X2 − p3TpSpX + p6S2

p

Recall the following formulae (see [A2]):

• T 2p − T (p2)− p2Sp = pRp + p(p2 + 1)Sp = (p + 1)−1(pT 2

p − pTp2 − (p4 − 1)Sp)

• pRp = T 2p − T (p2)− p(p2 + p+ 1)Sp

• pTp2 = (p+ 1)T (p2) + p2(p+ 1)Sp − T 2p

Also recall that S~n is a direct sum,⊕π, of irreducible admissible representations π =

⊗πl of GSp4(Af ). If p 6 |N and if πU 6= (0) then πp is spherical and so is the unique spherical

irreducible subquotient of some unramified principal series representation. (See [C] for this

and the facts quoted below about such representations.)

We first discuss the action of Hecke operators on unramified principal series representa-

tions, and then we apply these results to spaces of cusp forms. Fix the Borel:

B =

∗ 0 ∗ ∗∗ ∗ ∗ ∗0 0 ∗ ∗0 0 0 ∗

and the maximal torus T consisting of diagonal matrices. Describe unramified characters

on T (Qp) by triples (χ1, χ2, ψ) of unramified characters on Q×p , where:

(χ1, χ2, ψ) : diag(λ, µ, νλ−1, νµ−1) 7−→ χ1(λ)χ2(µ)ψ(ν)

In particular let δ denote the character taking this matrix to |λ2µ4ν−3|p. By the unramified

principal series corresponding to (χ1, χ2, ψ) we mean the representation on the space of

locally constant functions θ : GSp4(Qp) → C satisfying θ(bh) = ((χ1, χ2, ψ)δ12 )(b)θ(h) for

all b ∈ B(Qp) and h ∈ GSp4(Qp); where the action is given by (θg)(h) = θ(hg−1). We

shall denote this representation by π(χ1, χ2, ψ) and its irreducible spherical subquotient

37

by σ(χ1, χ2, ψ). Note this action is twisted from that in [C]. Then it is easy to see that

π(χ1, χ2, ψ)GSp4(Zp) = CΘχ1,χ2,ψ where:

Θ(bk) = Θχ1,χ2,ψ(bk) = ((χ1, χ2, ψ)δ12 )(b)

for b ∈ B(Qp) and k ∈ GSp4(Zp). That this is a good definition follows from the Iwasawa

decomposition. Then we can compute that:

• Θ|Tp = p32ψ(p−1)(1 + χ1(p

−1) + χ2(p−1) + χ1χ2(p

−1))Θ

• Θ|Sp = ψ(p−2)χ1χ2(p−1)Θ

• Θ|Rp = p2ψ(p−2)(χ1(p−1) + χ2(p

−1) + χ1χ2(p−1) + χ2

1χ2(p−1) + χ1χ

22(p−1))Θ

− ψ(p−2)χ1χ2(p−1)Θ

These follow from the coset decompositions (see [A2]):

• GSp4(Zp)diag(1, 1, p, p)GSp4(Zp) =∐GSp4(Zp)α as α runs over the matrices:

–

1 0 x y

0 1 y z

0 0 p 0

0 0 0 p

for x, y, z = 0, ..., p − 1

–

p 0 0 0

−i 1 0 z

0 0 1 i

0 0 0 p

for i, z = 0, ..., p − 1

–

1 0 x 0

0 p 0 0

0 0 p 0

0 0 0 1

for x = 0, ..., p − 1

38

–

p 0 0 0

0 p 0 0

0 0 1 0

0 0 0 1

• GSp4(Zp)diag(p, p, p, p)GSp4(Zp) = GSp4(Zp)diag(p, p, p, p)

• GSp4(Zp)diag(1, p, p2, p)GSp4(Zp) =∐GSp4(Zp)β as β runs over the matrices:

–

1 0 x y

0 p py 0

0 0 p2 0

0 0 0 p

for x = 0, ..., p2 − 1 and y = 0, ..., p − 1

–

p 0 0 py

i 1 y z

0 0 p pi

0 0 0 p2

for i, y = 0, ..., p − 1 and z = 0, ..., p2 − 1

–

p 0 0 0

0 p2 0 0

0 0 p 0

0 0 0 1

–

p2 0 0 0

−pi p 0 0

0 0 1 i

0 0 0 p

for i = 0, ..., p − 1

–

p12 B

0 p12

where B runs over non-zero symmetric 2 × 2 integral matrices

modulo p which satisfy detB ≡ 0 mod p

Then we easily conclude that:

39

• Θ|(pRp + p(p2 + 1)Sp) = p3ψ(p−2)(χ1(p−1) + χ2(p

−1) + 2χ1χ2(p−1) + χ2

1χ2(p−1) +

χ1χ22(p−1))Θ

• Θ|Qp(X) = (X−p 32ψ(p−1))(X−p 3

2ψχ1(p−1))(X−p 3

2ψχ2(p−1))(X−p 3

2ψχ1χ2(p−1))Θ

Now let Γ denote the subgroup of elements of GSp4(Zp) which are congruent to a matrix

of the form

∗ ∗

0 ∗

modulo p. Then GSp4(Qp) =

∐41B(Qp)wiΓ, where:

w1 = 14 w3 =

1 0 0 0

0 0 0 1

0 0 1 0

0 −1 0 0

w2 =

0 0 1 0

0 1 0 0

−1 0 0 0

0 0 0 1

w4 =

0 0 1 0

0 0 0 1

−1 0 0 0

0 −1 0 0

Thus π(χ1, χ2, ψ)Γ has a basis consisting of functions f1, f2, f3, f4 where fi is supported on

B(Qp)wiΓ and where fi(wi) = 1. We shall represent the function∑µifi by the row vector

(µ1, ..., µ4), so in particular Θ is represented by (1, 1, 1, 1). We shall calculate the matrix

representing the action of the Hecke operator Up = [Γdiag(1, 1, p, p)Γ] with respect to this

basis. It is easy to see that it is represented by (fj |Up(wi)), and we claim that this is:

p12ψ(p−1)

p p− 1 p− 1 p− 1

0 pχ1(p−1) (p− 1)χ1(p

−1) (p− 1)χ1(p−1)

0 0 pχ2(p−1) (p− 1)χ2(p

−1)

0 0 0 pχ1χ2(p−1)

To see this first note that:

(fj|Up)(wi) =∑

X

χ1χ2ψ2(p−1)fj

wi

p12 X

0 12

40

where X runs over 2× 2 integral matrices modulo p. To calculate these values we write:

wi

p12 X

0 12

= b(i,X)wk(i,X)γ(i,X)

where b ∈ B(Qp), γ ∈ Γ and k = 1, 2, 3 or 4. Then:

fj

wi

p12 X

0 12

=

((χ1, χ2, ψ)δ12 )(b(i,X)) if j = k(i,X)

0 otherwise

The b’s, γ’s and k’s are given by the following formulae:

• w1

p12 X

0 12

=

p12 X

0 12

w1

• If x 6≡ 0 mod p, say ax ≡ 1 mod p then:

w2

p 0 x y

0 p y z

0 0 1 0

0 0 0 1

=

p 0 −a −ay0 p −ay z − ay2

0 0 1 0

0 0 0 1

w1

−a 0 1p(1− ax) 0

ay 1 yp(1− ax) 0

−p 0 −x y

0 0 0 1

• w2

p 0 0 y

0 p y z

0 0 1 0

0 0 0 1

=

1 0 0 0

y p 0 z

0 0 p −y0 0 0 1

• If z 6≡ 0 mod p, say az ≡ 1 mod p, then:

w3

p 0 x y

0 p y z

0 0 1 0

0 0 0 1

=

p 0 x− ay2 −ay0 p −ay −a0 0 1 0

0 0 0 1

w1

1 −ay 0 yp(1− az)

0 −a 0 1p(1− az)

0 0 1 0

0 −p −y −z

41

• If y 6≡ 0 mod p, say ay ≡ 1 mod p, then:

w3

p 0 x y

0 p y 0

0 0 1 0

0 0 0 1

=

1 0 0 0

a p 0 a2x

0 0 p −a0 0 0 1

w2

0 a 1p(ay − 1) 0

−a a2x axp (ay − 1) 1

p(1− ay)p 0 x y

0 −p −y 0

• w3

p 0 x 0

0 p 0 0

0 0 1 0

0 0 0 1

=

p 0 x 0

0 1 0 0

0 0 1 0

0 0 0 p

w3

• If X is invertible modulo p, say Y X ≡ 12 mod p then:

w4

p12 X

0 12

=

p12 −Y

0 12

w1

−Y

1p(12 − Y X)

−p −X

• If xz = y2 and z 6≡ 0 mod p, say az ≡ 1 mod p, then:

w4

p 0 x y

0 p y z

0 0 1 0

0 0 0 1

=

1 0 0 0

−az p 0 a

0 0 p az

0 0 0 1

w2

1 ya 1p(ay

2 − x) yp(1− az)

0 a 0 1p(1− az)

0 0 1 0

0 −p y z

• If x 6≡ 0 mod p, say ax ≡ 1 mod p, then:

w4

p 0 x 0

0 p 0 0

0 0 1 0

0 0 0 1

=

p 0 a 0

0 1 0 0

0 0 1 0

0 0 0 1

w3

a 0 1p(1− ax) 0

0 1 0 0

−p 0 x 0

0 0 0 1

• w4

p12 0

0 12

=

12 0

0 p12

w4

42

From these results an easy calculation gives the matrix for the action of Up. Note that in

particular the eigenvalues of Up on π(χ1, χ2, ψ)Γ are p

32ψ(p−1), p

32χ1ψ(p

−1), p32χ2ψ(p

−1)

and p32χ1χ2ψ(p

−1).

Finally we are in a position to deduce the main result of this section:

Lemma 2.4 Let f ∈ S~n(U), where U is an open compact subgroup of∏GSp4(Zl) consist-

ing of all matrices congruent to

12 0

0 µ12

modulo some integer N ; be an eigenvector

of the Hecke operators Sl, Tl and T (l2) for all primes l 6 |N , say f |T = λ(T )f . Fix a prime

p 6 |N . Assume that λ(Qp(X)) has distinct roots the ratio of no two of which is p. Then if α

is a root of λ(Qp(X)) we can find a non-zero form f ′ ∈ S~n(U ∩U0(p)) which is an eigenform

of the Hecke operators Sl, Tl and T (l2) for l 6 |Np with eigenvalues given by λ, and such that

f ′|Up = αf ′. In particular if α is a p-adic unit then ef ′ = f ′.

Proof:We may write S~n =⊕π with π =

⊗πl being irreducible admissible representa-

tions of G(Af ). We may assume without loss of generality that f ∈ πU for some π. Then

πp is the spherical subquotient of an unramified principle series representation π(χ1, χ2, ψ),

where p 32ψ(p−1), p

32ψχ1(p

−1), p32ψχ2(p

−1), p32ψχ1χ2(p

−1), is the set of roots of λ(Qp(X)).

By our assumption that λ(Qp(X)) has distinct roots, (χ1, χ2, ψ) is regular (i.e. its conju-

gates under the Weyl group are distinct, i.e. χ1 6= χ2; χ1, χ2, χ1χ2 6= 1) and satisfies the

condition of theorem 3.10 of [C], namely:

(χ1, χ2, ψ)

(χ−11 , χ−12 , ψχ1χ2)

diag(p−1, p, p, p−1)

diag(p, 1, p−1, 1)

diag(1, p, 1, p−1)

diag(p, p, p−1, p−1)

6= p

i.e. χ2χ−11 (p) 6= p±1, χ1(p) 6= p±1, χ2(p) 6= p±1 and χ2χ1(p) 6= p±1. Thus π(χ1, χ2, ψ) is

irreducible and so the result now follows from the above discussion.

Remark: If λ(Qp(X)) has two distinct roots which are p-adic units and if 2 ≤ n1 ≤ n2

then it certainly satisfies the condition of the lemma.

43

Before finishing this section we give one further computation. If H is the Hecke algebra

of double cosets in GSp4(Zp)\GSp4(Qp)/GSp4(Zp) and if H′ is the Hecke algebra of double

cosets in GL2(Zp)\GL2(Qp)/GL2(Zp) then there is an injection: H → H′[x±1] (see for

example [Fa2]). An easy calculation shows that this map is given explicitly by:

[GSp4(Zp)gGSp4(Zp)] 7−→ xvalν (g)∑

GL2(Zp)ai

where GSp4(Zp)gGSp4(Zp) =∐GSp4(Zp)

ai bi

0 ν(g)ta−1i

. Thus if we let tp denote

[GL2(Zp)

1 0

0 p

GL2(Zp)] and sp denote [GL2(Zp)p12GL2(Zp)] we have that:

• Tp 7−→ (p3 + ptp + sp)x

• Sp 7−→ spx2

• Rp 7−→ (p3tp + sptp + (p2 − 1)sp)x2

• Qp(X) 7−→ (X − spx)(X − p3x)(X2 − ptpxX + p3spx2)

These follow from the decompositions given above. In particular we have that Qp(spx) = 0.

2.5 Main Results

We continue to assume that g = 2. We fix an integer N and we fix U to be the subgroup

of∏GSp4(Zl) consisting of matrices congruent to

12 0

0 12

modulo N . We fix also a

prime p not dividing N . We shall let U ′ denote any subgroup of the form:

x ∈ U

∣∣∣∣∣∣x ≡

αi ∗

0 λtα−1i

mod pr for some i

where r ≥ 1 and αi ⊂ GL2(Z/prZ) is some set. Also let TM denote the double coset

algebra over Z generated by the Hecke operators Tl, Sl and T (l2) for all primes l 6 |M . Then

we have:

44

Proposition 2.2 Let U ′ be as above and 3 ≤ n1 ≤ n2. Then there is a constant C such

that if f ∈ eS~n+m~t(U ′) (recall that ~t = (1, 1)) is an eigenform of the ring of Hecke operators

TNp, say f |T = λ(T )f ; then there is an integer a ≥ 1, a finite extension E/Qp such that

a[E : Qp] < C, and a continuous representation:

ρ : Gal(Qac/Q) −→ GLa(OE)

such that ρ is unramified outside Np and such that if l 6 |Np is a prime then λ(Ql)(Frobl) = 0.

Proof:We need only construct a representation valued in GLa(E), because as ρ is con-

tinuous and Gal(Qac/Q) is compact ρ will stabilise some compact OE module spanning Ea,

which then must be free.

In section 2.3 we define a representation V (~n + (m − 3)~t) of GSp4/Z. This gives us a

locally constant etale p-adic sheaf Vp(~n+(m−3)~t) on a certain smooth modelMM/Z[ 1M ] of

Z/ΓM , whereM = Npr+1 and ΓM denotes the set of matrices in Sp4(Z) which are congruent

to one modulo M . (See [Fa2].) Let W denote H3et(MM × specQ,Vp(~n+ (m− 3)~t)). Then

TNp, Up and Gal(Qac/Q) all act on W and these actions commute with one another. The

action of Gal(Qac/Q) is unramified outside Np and if l 6 |Np then Ql(Frobl) = 0. For these

assertions we refer the reader to [Fa2]. The operator Up is not treated there, but can be

treated in an exactly analogous manner. Note also that theorem one of the section “Hecke

Operators and Frobenius” in [Fa2] implies that Ql(Frobl) = 0 because we have seen at the

end of section 2.4 that Ql(xsl) = 0.

Now consider (eW )⊗TNpE, where E is the finite extension of Qp generated by the image

of λ and has a TNp action via λ. We must show that 1 ≤ dimQp(eW )⊗TNpE ≤ C, for C some

constant independent ofm and f . However (eW )⊗TNpE∼= eH3(ΓM , V (~n+(m−3)~t))⊗TNpE

and so dimQp(eW ) ⊗TNp E ≤ dimQp eW ≤ C for some constant C as in theorem 1.2.

Moreover if we fix E → C compatible with the p-adic valuation on Qac ⊂ C and if we let

C(λ) denote the one dimensional complex TNp module with the action via λ, then we see

that:

((eW )⊗TNp E)⊗ C ∼= eH3(ΓM , V (~n+ (m− 3)~t))⊗TNp C(λ)

⊃ eS~n(U ′)⊗TNp C(λ) 6= (0)

45

and we are done.

The following conjectural refinements of this proposition are well known:

Conjecture 2.1 Let U ′ be as above and 3 ≤ n1 ≤ n2. Then there is a constant C such

that if f ∈ eS~n+m~t(U ′) is an eigenform of the ring of Hecke operators TNp, say f |T =

λ(T )f ; then there is a finite extension E/Qp such that [E : Qp] < C, and a continuous

representation:

ρ : Gal(Qac/Q) −→ GSp4(OE)

such that ρ is unramified outside Np and such that if l 6 |Np is a prime then λ(Ql)(Frobl) = 0.

Conjecture 2.2 Let U ′ be as above and 3 ≤ n1 ≤ n2. Then there is a constant C such

that if f ∈ eS~n+m~t(U ′) is an eigenform of the ring of Hecke operators TNp, say f |T =

λ(T )f ; then there is a finite extension E/Qp such that [E : Qp] < C, and a continuous

representation:

ρ : Gal(Qac/Q) −→ GSp4(OE)

such that ρ is unramified outside Np and such that if l 6 |Np is a prime then Frobl has

characteristic polynomial λ(Ql).

Now we can state the main result of this chapter:

Theorem 2.2 Let U and U ′ be as above and let 2 ≤ n1 ≤ n2. Let χ : ((Z/Npr)×)2 → C×

be a character. Assume one of the following is true:

1. f ∈ eS~n(U ′)χ is an eigenform of TNp, say f |T = λ(T )f

2. f ∈ S~n(U)χ is an eigenform of TN , say f |T = λ(T )f , and λ(Qp(X)) has distinct

roots no quotient of two of which equals p and one of which is a p-adic unit.

3. f ∈ S~n(U)χ is an eigenform of TN , say f |T = λ(T )f , and λ(Qp(X)) has two distinct

roots which are p-adic units

Then there is a finite extension E/Qp, an integer a ≥ 1 and a continuous representation:

ρ : Gal(Qac/Q) −→ GLa(OE)

46

which is unramified outside Np and such that if l 6 |Np is a prime then λ(Ql)(Frobl) = 0.

Proof:By lemma 2.4 of section 2.4 the second two cases reduce at once to the first case.

So consider the first case. We may find forms fm ∈ eS~n+bm~t(U′)χ which are eigenvalues of

TNp, say fm|T = λm(T )f , and such that as m goes to infinity bm →∞ and supTNp |λm(T )−λ(T )|p → 0. (Lemma 2.1 of section 2.3.) Then for any positive integer s we may find a

positive integer as, a finite extension Es/Qp with as[Es : Qp] < C (where C is as in the last

proposition, but for (n1 + 1, n2 + 1)) and a continuous representation ρs : Gal(Qac/Q) →GLas(OEs/ps) which is unramified outside Np and such that λ(Ql)(Frobl) = 0. However

there are only finitely many extensions E : Qp of degree less than C, so we may assume

that Es = E and as = a are independent of s.

We shall recursively define infinite subsets It ⊂ It−1 ⊂ T such that if s1, s2 ∈ It thenρs1 ≡ ρs2 mod pt. This is possible as (ρs mod pt) factors through Gal(K/Q) for some

Galois extension K/Q unramified outside Np and of degree bounded independently of s. It

is known that there are only finitely many such extensions and so there is a finite Galois

extension L/Q through which all the (ρs mod pt) factor. Now there are only finitely many

maps Gal(L/Q)→ GLa(OE/pt) so for infinitely many s ∈ It−1 the maps (ρs mod pt) must

be equal as desired. Now we define ρ : Gal(Qac/Q) → GLa(OE) by the requirement that

ρ(σ) ≡ ρs(σ) mod pt for all σ ∈ Gal(Qac/Q) and all t and all s ∈ It. This is easily checked

to be a good definition of a morphism with the desired properties.

If we assume the conjectures mentioned above the same method gives the following

conjectural strengthening of this theorem:

Theorem 2.3 Assume conjecture 2.1 (resp. 2.2). Let U and U ′ be as above and let 2 ≤n1 ≤ n2. Let χ : ((Z/Npr)×)2 → C× be a character. Assume one of the following is true:

1. f ∈ eS~n(U ′)χ is an eigenform of TNp, say f |T = λ(T )f

2. f ∈ S~n(U)χ is an eigenform of TN , say f |T = λ(T )f , and λ(Qp(X)) has distinct

roots no quotient of two of which equals p and one of which is a p-adic unit.

47

3. f ∈ S~n(U)χ is an eigenform of TN , say f |T = λ(T )f , and λ(Qp(X)) has two distinct

roots which are p-adic units

Then there is a finite extension E/Qp and a continuous representation:

ρ : Gal(Qac/Q) −→ GSp4(OE)

which is unramified outside Np and such that if l 6 |Np is a prime then λ(Ql)(Frobl) = 0

(resp. Frobl has characteristic polynomial λ(Ql(X))).

48

Chapter 3

p-adic Families of Siegel Modular

Forms

3.1 Introduction

In this chapter we develop a theory of Hida families for Siegel modular forms of even genus.

To explain our results fix an odd prime p, an even positive integer g and embeddings

Qac ⊂ C, Qac ⊂ Qacp . The assumption that g be even is probably not essential. It is made

because our computations with Eisenstein series and Rankin’s method are dependent on the

parity of g, and because g = 2 is the case of greatest interest for us. Let Mk(N,χ) denote

the space of Siegel modular forms of genus g, weight k, level N and character χ. This

space has a Z[ζ] structure for some root of unity ζ and this allows us to define Mk(N,χ,A)

for any algebra A containing enough roots of unity so that we can consider χ as valued in

A. Any element of Mk(N,χ,A) has a formal Fourier expansion∑ah exp(πi tr(hz)) with

ah ∈ A and where h runs over the set S of integral, positive semi-definite g × g symmetric

matrices with even diagonal entries. If p|N we shall let Up denote the Hecke operator

[Γ0(N)

1g 0

0 p1g

Γ0(N)] where Γ0(N) is defined in section 3.2. If O denotes the integers

of a suitably large extension of Qp we define Mk (N,χ,O) to be the largest submodule of

Mk(N,χ,O) on which Up is an automorphism. It is in fact a direct summand, which we

49

shall call the ordinary part. We can carry this notion over to other settings, for example

we can define Mk (N,χ,C) using the embeddings Qac ⊂ C and Qac ⊂ Qacp . Note that if

f ∈ Mk(N,χ) is an eigenform of Up, say f |Up = apf , then f ∈ Mk (N,χ) if and only if ap

is a p-adic unit. We shall also define in section 3.2 a certain commutative ring of Hecke

operators TN which acts semi-simply on Mk(N,χ) and commutes with the action of Up.

Now let N be prime to p. Let Λ be the power series ring O[[T ]] with O as above. By

a Λ-adic form of level N and character χ (defined modulo Np), we shall mean a formal

expansion:∑

h∈S

ahqh

with coefficients in Λ and such that for all but finitely many pairs (k, α) with k ≥ g+1 and:

α : (Z/Np?Z)× → (1 + pZ)/(1 + p?Z)→ Qac×

we have that:∑

h∈S

ah(α(1 + p)(1 + p)k − 1) exp(πi tr(hz))

is the Fourier expansion of an element of Mk (Np?, χω−kα,Qac

p ). We shall denote the space

of such forms M(N,χ). We can define an action of TNp on M(N,χ) compatible with

specialisation.

Our first main theorem states thatM(N,χ) is a finite free Λ module. The main point of

the proof is that dimMk (Np,χ) is bounded independently of k. The second main theorem

states that if f ∈Mk (Np?, χω−kα) is an eigenvector of TNp, say f |T (n) = λ(n)f , then we

can find an integerM (divisible byN) and a form F ∈M(M,χ)⊗R (R the integers of some

extension of the field of fractions of Λ) which is an eigenvector for TMp, say F |T (n) = λ(n)F ,

such that λ(n) ≡ λ(n) modulo some prime of R lying above (1 + T − α(1 + p)(1 + p)k.

To prove this we follow a method of Wiles (see [Wi]). One first writes down some Λ-adic

Eisenstein series, one then multiplies them by a certain form of low weight (we use a theta

series) and uses Rankin’s method to show that if f is of high enough weight it will occur

in the spectral decomposition of this product. For this we must use our first theorem. To

50

extend the result to all weights k one shows that if f is an ordinary eigenform of weight k we

can find ordinary eigenforms arbitrarily congruent to f with weights of the form k+(p−1)p?.In the last section we show how to use this theory to rederive some results of the last

chapter about Galois representations. We also show how the standard conjectures on the

existence of Galois representations one expects to associate to Siegel modular forms would

imply similar results about Λ-adic representations.

3.2 Review of Siegel Modular Forms II

We set up a theory of Siegel modular forms in a more classical setting than the last chapter.

This is better suited to the purposes of this chapter.

We shall fix throughout an odd prime p and embeddings Qac → C and Qac → Qacp . We

shall also fix a positive integer g. For our main results we must assume that g is even, and

we shall in fact always assume this except in sections one and four where we shall need to

consider all g to make a certain induction argument work.

We shall let Zg denote the Siegel space of genus g, that is:

Zg = x+ iy |x, y ∈ symmg(R), y > 0

Here we write a > 0 if a is a positive definite symmetric real matrix. Also if A is a ring

symmg(A) denotes the module of g × g symmetric matrices over A. Moreover symm∗g(A)

will denote the sub-module whose diagonal entries are in 2A and we shall use the superscript

≥ 0 to denote the sub-semigroup of positive semi-definite elements when this makes sense.

A couple more notes on notation. We shall use ǫij to denote the g × g matrix that has

one at the intersection of the ith row and jth column and zeroes elsewhere, and 1g to denote

the g × g identity matrix. We shall let ν : GSp2g → Gm denote the character such thatν

A B

C D

1g = AtD−BtC. Also if R is an integral domain FR will denote its field

of fractions.

51

The group GSp+2g(R) acts on Zg by:

A B

C D

: z 7−→ (Az +B)(Cz +D)−1

If f : Zg → C is a function, k is a non-negative integer and γ ∈ GSp+2g(R) we define a

transform of f by γ by the formulae:

• f |kγ(z) = ν(γ)gk2 j(γ, z)−kf(γz)

• j

A B

C D

, z

= det(Cz +D)

If Γ is a subgroup of Sp2g(Z) of finite index we define a space of modular forms, denoted

Mk(Γ), to be the set of holomorphic functions f : Zg → C such that f |kγ = f for all γ ∈ Γ.

If g = 1 we must supplement these conditions with a growth condition, but this is well

known. If we wish to indicate the genus we shall write M(g)k (Γ). In this work we shall be

concerned with two special congruence subgroups of Sp2g(Z) which we shall denote:

• Γ0(N) =

A B

C D

∈ Sp2g(Z) |C ≡ 0 mod N

• Γ1(N) =

A B

C D

∈ Sp2g(Z) |C ≡ 0 mod N, detD ≡ 1 mod N

We shall decompose:

Mk(Γ1(N) =⊕

χ

Mk(N,χ)

as χ runs over characters χ : (Z/NZ)× → Qac×, and where Mk(N,χ) denotes the space

of holomorphic functions f : Zg → C such that f |kγ = χ(γ)f for all γ ∈ Γ0(N). Here

χ : Γ0(N)→ Qac× by

A B

C D

7→ χ(detD). Again we need a growth condition if g = 1.

Any element of Mk(Γ1(N)) has a Fourier expansion:

∑ah(f) exp(πi tr(hz))

52

as h runs over symm∗g(Z)≥0. If R is a ring we shall denote by R[[q]]g the ring of formal

power series∑ahq

h as h runs over the semigroup symm∗g(Z)≥0, and where ah ∈ R. With

this notation we see that:

Mk(Γ1(N)) → C[[q]]g

f 7−→ ∑ah(f)q

h

If R ⊂ C is a sub-Z-module we define Mk(Γ1(N), R) to be those elements of Mk(Γ1(N))

whose Fourier expansion lies in R[[q]]g ⊂ C[[q]]g. It is a result of Shimura thatMk(Γ1(N),C)

=Mk(Γ1(N),Z)⊗ZC and so for any Z-moduleR we may consistently defineMk(Γ1(N), R) =

Mk(Γ1(N),Z) ⊗Z R. Similarly if Oχ denotes the extension of Z generated by the im-

age of χ then Mk(N,χ,Oχ) ⊗Oχ C = Mk(N,χ) so if R is any Oχ module we can define

Mk(N,χ,R) = Mk(N,χ,Oχ) ⊗Oχ R. Note also that this implies that Aut(C/Q) acts on

Mk(Γ1(N)) by:

fσ(z) =∑

ah(f)σ exp(πi tr(hz))

where σ ∈ Aut(C/Q). In fact Mk(N,χ)σ =Mk(N,σ χ).

If f is a modular form of genus g for some congruence subgroup Γ of Sp2g(Z) we define

a modular form Φ(f) of genus g − 1 by:

Φ(f)(z) = limλ→+∞

f

z 0

0 iλ

Φ is called the Siegel operator. Its action on Fourier expansions is given by:

Φ(f) =∑

ah′(f) exp(πi tr(hz))

where h′ =

h 0

0 1

. We can embed Sp2(g−1) into Sp2g by:

A B

C D

7−→

A B

1 0

C D

0 1

53

If Γ is a congruence subgroup of Sp2g set Φ(Γ) = Γ ∩ Sp2(g−1). Then:

Φ :M(g)k (Γ) −→M

(g−1)k (Φ(Γ))

and:

Φ :M(g)k (N,χ) −→M

(g−1)k (N,χ)

We call a modular form of weight k cuspidal if for all γ ∈ Sp2g(Z) we have that Φ(f |kγ) =0. We denote the subspace of Mk(Γ) consisting of cusp forms by Sk(Γ). Similarly we use

the notation Sk(N,χ). The rationality results described above remain true for cusp forms.

If Sp2g(Z) = ∐IΓδi then we have a left exact sequence:

0→ S(g)k (Γ)→M

(g)k (Γ)→

⊕

I

M(g−1)k (Φ(δ−1i Γδi))

and similarly for Mk(N,χ).

If f ∈ Sk(Γ) and g ∈Mk(Γ) we can define an inner product 〈f, g〉Γ by:∫

Γ\Zg

f(z)g(z)(det y)k dz

where z = x + iy and dz = (det y)−g−1∏

1≤α≤β≤g dxαβ dyαβ. Then if γ ∈ GSp+2g(R) we

have that 〈f |kγ, g|kγ〉γ−1Γγ = 〈f, g〉Γ. Similarly if f ∈ Sk(N,χ) and g ∈Mk(N,χ) we set:

〈f, g〉N =

∫

Γ0(N)\Zg

f(z)g(z)(det y)k dz

If U denotes an operator on Sk(N,χ) we shall let U∗ denote its adjoint with respect to

this inner product. It will be convenient to introduce a slight variant variant of this inner

product. Set:

WN =

0 −1g

N1g 0

then for f ∈ Sk(N,χ) and g ∈Mk(N,χ) we define:

(f, g)N = 〈f |WN , gc〉N

where c denotes complex conjugation. This is a C bilinear form which restricted to Sk(N,χ)2

is non-degenerate. If U is an operator on Sk(N,χ) we shall let U† denote its transpose with

respect to this pairing.

54

We shall now recall some facts about the theory of Hecke operators (see [A2] for details).

If g ∈ GSp+2g(Q) and g ≡

∗ ∗

0 ∗

mod N we define the Hecke operator:

[Γ0(N)gΓ0(N)] :Mk(N,χ) −→Mk(N,χ)

by:

f 7−→ ν(g)gk2−g(g+1)

2

∑

I

χ′(gi)f |kgi

where:

• Γ0(N)gΓ0(N) = ∐Γ0(N)gi

• χ′ A B

NC D

= χ(detA) this being given the value 0 if (detA,N) 6= 1.

Lemma 3.1 If f1 ∈ Sk(N,χ), f2 ∈Mk(N,χ), g =

A B

NC D

∈ GSp+2g(Q) and

A,B,C,D ∈Mg×g(Z) then:

〈(f1|[Γ0(N)gΓ0(N)])|WN , f2〉N = 〈f1|WN , f2|[Γ0(N)g∗Γ0(N)]〉

where:

g∗ =

N−11g 0

0 1g

tg

N1g 0

0 1g

=

tA tC

N tB tD

In particular if g is as above and if further:

f c|[Γ0(N)gΓ0(N)] = (f |[Γ0(N)gΓ0(N)])c

then [Γ0(N)gΓ0(N)]† = [Γ0(N)gΓ0(N)].

55

Proof:Let Γ0(N)gΓ0(N) = ∐Γ0(N)gγi with γi =

Ai Bi

Ci Di

∈ Γ0(N). Then:

〈(f1|[Γ0(N)gΓ0(N)])|WN , f2〉N

=∑

i ν(gγi)?χ(AAi)[Γ0(N) : Γ0(N) ∩ γ−1i g−1Γ0(N)gγi]

−1

〈f1|WN (W−1N gγiWN ), f2〉Γ0(N)∩γ−1

i g−1Γ0(N)gγi

= ν(g)?χ(A)[Γ0(N):Γ0(N)∩g−1Γ0(N)g]

∑i〈f1|WN (W

−1N (g∗)−1, χ(Di)f2|γ∗i 〉Γ0(N)∩g−1Γ0(N)g

= ν(g)?χ(A)〈f1|WN (g∗)−1, f2〉Γ0(N)∩g−1Γ0(N)g

where ? = gk2 −

g(g+1)2 . Similarly:

〈f1|WN , f2|[Γ0(N)g∗Γ0(N)]〉N = ν(g∗)?χ(A)〈f1|WN , f2|g∗〉g∗−1Γ0(N)g∗∩Γ0(N)

and so the lemma follows.

Let ∆0(N) be the semigroup of elements γ in GSp+2g(Q) ∩M2g(Z) with (N,det γ) = 1

and γ ≡

∗ ∗

0 ∗

mod N . Let T∗N denote the Hecke algebra which is spanned over Z

by the double Γ0(N) cosets contained in ∆0(N). Then T∗N acts on Mk(N,χ) and this

action preserves Sk(N,χ). Moreover if N |M then T∗M → T∗N and if N and M have the

same prime factors this is an isomorphism. This map is also compatible with the inclusion

Mk(N,χ) → Mk(M,χ). Any element T of T∗N can be written as∑niΓ0(N)gi with gi =

Ai Bi

0 Di

and then:

ah(f |T ) =∑

niν(gi)−g(g+1)

2 χ(detAi)(detAi)k exp(πi tr(hA−1i Bi))aDihA−1

i(f)

where the sum is taken over those i such that DihA−1i ∈ symm∗g(Z). In particular T∗N

preserves Mk(N,χ,Qac). It is also known that Sk(N,χ) has a basis of eigenforms for T∗N ,

which are orthogonal with respect to the Petersson inner product. We see that:

• If M ⊂Mk(N,χ) is preserved by T∗N so is M⊥ ⊂ Sk(N,χ)

• If U is an operator on Sk(N,χ) which commutes with the action of T∗N then U∗ also

commutes with T∗N .

56

For n a positive integer prime to N let T (n) ∈ T∗N denote the sum of all Γ0(N) double

cosets in ∆0(N) on which ν takes the value n. We let TN denote the subring of T∗N generated

by these operators. We have that:

ah(f |T (n)) = n−g(g+1)

2 χ(ng)ngk∑

d1|d2|...dg|adg1d

g−12 ...dg

∑D χ(detD)−1(detD)−kan−1DhtD

where the second sum is taken over a set of representatives for:

GLg(Z)\GLg(Z)

d1. . .

dg

GLg(Z)

In particular TN preservesMk(N,χ,Oχ,℘) for ℘ any prime of Oχ dividingN . Also if T ∈ TN

then T † = T . This follows from lemma 3.1 and the fact that ∆0(N)∗ = ∆0(N).

For all these results we refer the reader to [A2]. We also note that any eigenform of T∗N

is also an eigenform of each algebra Lg0,l(N) (l 6 |N) considered in [A1] (this follows from [A2]

theorem 4.1.8).

If p|N we shall also consider the operators Upr = [Γ0(N)

1g 0

0 pr1g

Γ0(N)]. We list

some properties of Upr :

Lemma 3.2 Let p|N and let χ be defined modulo N . Then Up ∈ End(Mk(N,χ)) satisfies:

1. if N |M then the action of Up is compatible with Mk(N,χ) →Mk(M,χ)

2. ah(f |Upr) = aprh(f)

3. Upr = U rp

4. UpMk(Np,χ) ⊂Mk(N,χ)

5. Up commutes with the action of TN

6. U †p = Up

7. if T ∈ T∗N then some power of Up commutes with T .

57

Proof:1. and 2. follow from the fact that:

Γ0(N)

1g 0

0 pr1g

Γ0(N) = ∐Γ0(N)

1g B

0 pr1g

where B runs over any set of representatives for the mod pr congruence classes of symmg(Z).

3) follows from 2). 4) follows from the fact that:

Γ0(Np)

1g 0

0 p1g

Γ0(N) = ∐Γ0(Np)

1g B

0 p1g

where B is as above. 5) follows from 2) and the formula for ah(f |T (n)), and 6) follows

from 2) and lemma 3.1. We do not prove these coset decompositions here. The proof is

essentially the same as that given in lemma 3.4 below.

7) follows from these decompositions and the facts that if

A B

0 D

∈ ∆0(N), µ =

ν(

A B

0 D

), X ≡ 0 mod µ is a symmetric integral matrix, and pr ≡ 1 mod µ then:

1g X

0 pr1g

A B

0 D

=

A B

0 D

1g µ−1tD((1 − pr)C +XD)

0 pr1g

and that X 7→ µ−1tD((1 − pr)B +XD) is a permutation of the modpr congruence classes

of symmetric integral matrices.

Note that in particular Up preserves Mk(N,χ,Oχ).We now want to define the Hida idempotent associated to Up which will be essential in

what follows. First recall the following lemma:

Lemma 3.3 Let O denote the integers of a finite extension of Qp and M a finite O-module.

Let U be an operator on M , then there exists a unique idempotent eU in EndO(M) which

commutes with U and such that U is an automorphism on eM and is topologically nilpotent

on (1−e)M . Moreover e = limr→∞Ur!. IfM ′ and U ′ satisfy corresponding conditions and if

α :M →M ′ is such that αU = U ′α then αe = eα.

58

If O0 is a number field, M0 a finite O0-module, ℘ a prime of O0 above p and U ∈EndO0(M0) then there is a ring R contained in a number field with O0 ⊂ R ⊂ O0,℘ and an

idempotent eU ∈ EndR(M0 ⊗ R) such that eU considered as an element of EndO0,℘(M0 ⊗O0,℘) coincides with the idempotent associated to U above.

In particular in these circumstances we can think of e ∈ EndQac(M0 ⊗ Qac), as we have

fixed Qac ⊂ Qacp .

We shall be interested in the case M0 = Mk(N,χ,Oχ) with p|N and U = Up. We shall

denote by e the corresponding idempotent, which we shall call the Hida idempotent. We can

think of e acting on Mk(N,χ,Oχ,℘) or on Mk(N,χ). In either case we have the following

properties:

• If N |M the action of e is compatible with Mk(N,χ) →Mk(M,χ)

• eMk(Npr, χ) = eMk(N,χ)

• e commutes with the action of T∗N

• e† = e

Finally we must study the operator Up and the Hida idempotent e in a slightly different

setting. Let N now be an integer prime to p.Now let Γ denote one of the following:

• Γ0 =

A B

C D

∈ Sp2g(Z) |B ≡ 0 mod N, C ≡ 0 mod Np,

A ≡ D ≡ 1 mod N

• Γ1 =

A B

C D

∈ Γ0 | detD ≡ 1 mod p

• Γ1(Npr)

Any element of Mk(Γ) has a Fourier expansion∑ah(f)q

h where h runs over elements of

N−1symm∗g(Z). The theory of rationality carries over exactly to this situation. We define

59

operators Upr to be the Hecke operators associated to the double cosets Γ

1g 0

0 p1g

Γ.

Then we have:

Lemma 3.4 With the notation as above:

1. Upr = U rp

2. ah(f |Up) = aph(f)

3. Up preserves Mk(Γ,Z)

4. the action of Up is compatible with the inclusions:

Mk(Γ1) ⊃ Mk(Np,χ)

∪Mk(Γ0) ⊃ Mk(Np, 1)

and:

Mk(Γ1(Npr)) ⊃Mk(Np

r, χ)

Proof:These all follow from the facts that:

Γ

1g 0

0 p1g

Γ = ∐Γ

1g B

0 p1g

where B runs over a set of representatives for the mod pr congruence classes of symmg(Z)

each chosen ≡ 0 mod N . This decomposition follows from the following equations:

•

1g B

0 pr1g

=

1g 0

0 pr1g

1g B

0 1g

•

1g X

0 pr1g

1g Y

0 pr1g

−1

=

1g p−r(X − Y )

0 1g

60

• If

A B

pC D

∈ Sp2g(Z) then tDB ∈ symmg(Z) and if further X ∈ symmg(Z) with

X ≡t DB mod pr and X ≡ 0 mod N then:

1g 0

0 pr1g

A B

pC D

=

A p−r(B −AX)

pr+1C D − pCX

1g X

0 pr

where B −AX ≡ A(tDB −X) ≡ 0 mod pr.

In particular we can introduce the Hida idempotent in this context, and it will be com-

patible with the other contexts considered earlier. We finish this section with a simplification

of the condition that an ordinary modular form is cuspidal:

Lemma 3.5 Let N be prime to p. Let Γ (or Γ(g)) denote either Γ0 or Γ1 as defined above.

Let Γ(g−1) denote the corresponding group in genus g − 1. Then there is an integer m such

that for all k ≥ g(g+1)2 we have a left exact sequence:

0→ eSk(Γ(g))→ eMk(Γ

(g))→ eMk(Γ(g−1))m

Proof:Let Γ\Sp2g(Z) = ∐IΓδi, so that we have an exact sequence:

0→ Sk(Γ)→Mk(Γ)φ→

⊕

I

Mk(Φ(δ−1i Γδi))

Let J be the subset of I consisting of those i for which Γδi ⊂ Γ0(p). Let Ω ⊂ Qac be the

finite Z[ζp]-module generated by the coefficients f |δi where f ∈ Mk(Γ,Z) and i ∈ I. Then

we make the following claims:

1. f ∈Mk(Γ,Z) and φ(f) ∈⊕

I\J Mk(Φ(δ−1i Γδi), p

rΩ) implies that φ(f |Up) ∈⊕

IMk(Φ(δ−1i Γδi), p

r+1Ω)

2. f ∈Mk(Γ) and α ∈ Γ0(p) implies that α−1Γα = Γ and that f |α|Up = f |Up|α

3. f ∈Mk(Γ,Z) and φ(f) ∈⊕

I\J Mk(Φ(δ−1i Γδi), p

rΩ) implies that φ(f |Up) ∈⊕

I\JMk(Φ(δ−1i Γδi), p

(r + 1)Ω)

61

4. The following sequence is left exact:

0→ eSk(Γ)→ eMk(Γ)→⊕

J

Mk(Γ(g−1))

5. Φ :Mk(Γ(g))→Mk(Γ

(g−1)) is compatible with the action of Up.

Note that from 2), 4) and 5) the lemma follows at once. Also note that 5) follows easily

from the description of Up in terms of its action on Fourier expansions; and that 3) and 4)

follow easily from 1) and 2).

First we prove 1). Note that for α =

A B

C D

∈ Sp2g(Z):

f |Up|α = pgk2− g(g+1)

2

∑

X

f |

A+XC B +XD

pC pD

and that: A+XC B +XD

pC pD

=

(A+XC)E−1 B′

pCE−1 D′

E Y

0 ptE−1

with:

•

(A+XC)E−1 B′

pCE−1 D′

∈ Sp2g(Z)

•

E Y

0 ptE−1

∈M2g×2g(Z)

• E upper triangular

(That we can find such a decomposition can be easily deduced from proposition 1.3.7 of

[A2].) We now consider two cases:

1. p 6 | detE, i.e. E ∈ GLg(Z)In this case:

A+XC B +XD

pC pD

=

A+XC B′tE−1

pC D′tE−1

1g E−1Y

0 p1g

62

with

A+XC B′tE−1

pC D′tE−1

∈ Γ0(p). Thus:

Φ

f |

A+XC B +XD

pC pD

= 0

2. p|detEIn this case:

pgk2−g(g+1)

2 f |

A+XC B +XD

pC pD

=

pk−g(g+1)

2 (p−1 detE)k∑ah exp(πi tr(p

−1hY tE)) exp(πi tr(p−1 tEhEz))

where:

f |

(A+XC)E−1 B′

pCE−1 D′

=

∑ah exp(πi tr(hz))

In either case φ(f |Up) ∈⊕

IMk(Γ, pr+1Ω) as E upper triangular implies that

tE−1

h

0

E−1 =

h′

0

.

Finally for claim 2), let X ≡ 0 mod N and

A B

pC D

∈ Γ0(p). Then:

A B

pC D

1g X

0 p1g

A B

pC D

−1

1g AXtA−AtB0 p1g

−1

=

AtD −BtCp2 − pAXtC −AtB +BtAp +AXtA

p(CtD −DtCp− CXtCp) p(−CtB +DtA+ CXtA)

1g p−1(AtB −AXtA)

0 p−11g

• ∈ Sp2g(Z) as AtD(AtB −AXtA)−AtB +AXtA ≡ 0 mod p

• ≡

A′ ∗

0 D′

mod p and if detA ≡ detD ≡ 1 mod p then detA′ ≡ detD′ ≡ 1 mod

p.

63

• ≡

1g 0

0 1g

mod N

The result now follows as X 7→ AXtA−AtB is a permutation of symmg(Z/pZ).

3.3 Some Lemmas on Eisenstein and Theta Series

In this section we collect some results that we need concerning Eisenstein series, theta series

and Rankin’s method. These results are based on work of Andrianov and of Shimura. First

we introduce some Eisenstein series:

• E(z, s; k, φ, b) = (det 2y)s∑φ(detD) det(Cz+D)−k|det(Cz+D)|−2s where z = x+iy,

and where the sum runs over

A B

C D

representing the cosets Γ0\Γ0(b).

• E∗(z, s; k, φ, b) = E(−1z , s; k, φ, b)z−k

• Ek,φ,b(z) =∑

h ∈ symm∗g(Z)

h > 0

ah exp(πi tr(hz)) with:

ah = L(1 + g2 − k, φ−1ξh)

∏l ∈ P (h)

l 6 |b

Ml,h(φ(l)lk

)

(det h)k−

g+12 f

g2−k

φξh

τ(φξh)∏l 6 |fφξh

l | b

(1− φξh(l)lg2−k)

where

– ψ is the primitive character corresponding to ψ

– if ψ is a primitive character fψ denotes its conductor and

τ(ψ) =∑

x∈(Z/fψZ)×

ψ(x) exp(2πix

fψ)

denotes the corresponding Gauss sum

– ξh = ζhθg2 where ζh is the character corresponding to Q(

√det h)/Q and θ is the

character corresponding to Q(√−1)/Q

64

– P (h) is a finite set of primes depending only on Q×h, as defined in §10 of [Fe].

– for l ∈ P (h) Ml,h(X) ∈ Z[X] is a polynomial depending only on l and Q×h as

defined in §10 of [Fe]

Lemma 3.6 Assume k ≥ g2 + 1 and that φ is a character modulo b, with φ(−1) = (−1)k.

Assume also that either k > g2 + 2 or φ2 6= 1. Then:

CEk,φ,b(z) = E∗(z, 0; k, φ, b) | [b] ∈Mk(Γ0(b), φ)

with C 6= 0 a constant.

Proof:E∗(x+ iy, s; k, φ, b) =∑

bh∈symm∗g(Z)

ah(y, s) exp(πi tr(hx)) where

ah(y, s) has been calculated by Shimura and Feit. In particular ah(y, s) is finite (resp. zero)

at s = 0 if Ah(s) is finite (resp. zero) at s = 0, where, if h has r+ positive eigenvalues, r−

negative eigenvalues and rank r = r+ + r−:

Ah(s) =Γg−r(k−

g+12

+2s)

Γg−r−(k+s)Γg−r+ (s)

(L(k + 2s, φ)

∏ g2i=1 L(2k − 2i+ 4s, φ2)

)−1

.

L(k − g + r2 + 2s, φξh)

∏ g−r2

i=1 L(2k − 2g + r − 1 + 2i+ 4s, φ2)∏ g−r−1

2i=0 L(2k − 2g + r + 2i+ 4s, φ2)

according as r is even or odd. Here Γt(s) =∏t−1i=1 Γ(s − i

2). (See §10 of [Fe].) The only

terms that contribute a zero or pole at s = 0 are:

• Γg−r+(s)−1 which contributes a zero of order the integer part of g−r++1

2

• the L-series in the numerator which can contribute at most a simple pole.

Thus Ah(s) has a pole at s = 0 only if g = r+ and the L-series in the numerator have a

pole, i.e. k = 1+ g2 , φξh = 1, which is a case we have excluded. Moreover Ah(0) = 0 unless

g = r+ or one of the following hold:

• g = r+ + 1, k = g2 + 1, φ2 = 1

• g = r+ + 2, k = g2 + 2, φξh = 1

65

• g = r+ + 2, k = g2 + 1, φ2 = 1

However our assumptions exclude these last three possibilities. Thus we have:

E∗(x+ iy, 0; k, φ, b) =∑

h ∈ symm∗g(Z)

h > 0

ab−1h(y, 0) exp(πi tr(hx))

where using [Fe] §10 and the functional equation for L-series associated to Dirichlet char-

acters:

ah(y, o) = 2?π?i?(det b)?γL−1ahe−π tr(hy)

with each ? a rational number independent of h and with:

• γ = Γg(k)−1Γ

(k+µ− g

22

)−1Γ(1−k+µ+ g

22

)

• L = L(k, φ)∏ g

2i=1 L(2k − 2i, φ2)

• µ = 0 or 1 and µ ≡ k − g2 mod 2, so that φξh(−1) = (−1)µ.

From this the desired equality follows at once. Moreover we see that E∗(z, 0; k, φ, b) | [b]is holomorphic. Finally it lies in Mk(Γ0(b), φ) because it is equal to E(z, 0; k, φ, b) |Wb and

E(z, s; k, φ, b) transforms under Γ0(b) by φ−1.

We now introduce some theta series. Let ψ be a character modulo r and let Q ∈symm∗g(Z), Q > 0. We shall let s(Q), the step of Q, denote the smallest positive integer

such that s(Q)Q−1 ∈ symm∗g(Z). Now set:

θQ,ψ(z) =∑

N∈Mg×g(Z)

ψ(detN) exp(πi tr(tNQNz))

We first record some transformation properties of θQ,ψ. The proofs are based on work of

Andrianov. In fact part one is due to him in the case ψ primitive, but unfortunately we

don’t think this method goes over exactly to the case of ψ imprimitive.

Lemma 3.7 1. θQ,ψ(z) ∈ M g2(Γ0(r

2s(Q)), ψχQ) where χQ is a character of order two

modulo s(Q) with χQ(−1) = (−1) g2 .

66

2. If ψ is primitive modulo r then θQ,ψ |Wr2s(Q) = Cθs(Q)Q−1,ψ−1 for some constant C.

Proof:Following Andrianov we introduce for T ∈ Mg×g(Z) with QT ≡ 0 mod s(Q) a

theta series:

θ(z,Q : T ) =∑

N∈Mg×g(Z)

exp(πi tr(t(N + s(Q)−1T )Q(N + s(Q)−1T )z))

Then:

θQ,ψ(z) =∑

M∈Mg×g(Z)modr

ψ(detM)θ(z, r2Q : rs(Q)M)

Now from §1.3.3 of [A2] we know that:

1. If γ =

A B

C D

∈ Γ0(s(Q)) then:

θ(z,Q : T ) | g2γ = χQ(detD) exp(πi tr(s(Q)−2AtBtTQT ))θ(z,Q : TA)

2. θ(z,Q : T ) | Ws(Q) = C1∑

QU ≡ 0 mod s(Q)

U mod s(Q)

exp(2πi tr(s(Q)−2 tTQU))θ(z, s(Q)Q : U)

with C1 a non-zero constant independent of T .

Thus if γ =

A B

C D

∈ Γ0(r

2s(Q)) then:

θQ,ψ | g2γ(z) =

∑ψ(detM)χQ(detM) exp(πi tr(AtBtMQM))

θ(z, r2Q : rs(Q)MA)

= χQψ(detD)∑ψ(det(MA))θ(z, r2Q : rs(Q)MA)

= χQψ(detD)θQ,ψ(z)

(Here we have used the fact that (AtB)(tMQM) has trace an even integer.)

Now assume ψ is primitive. For A ∈Mg×g(Z) set:

τA(ψ) =∑

M∈Mg×g(Z/rZ)

ψ(detM) exp(2πi

rtr(MA))

We shall assume for the moment the following two facts:

67

• A ∈ GLg(Z/rZ) implies that τA(ψ) = ψ(detA)−1τ1(ψ)

• A 6∈ GLg(Z/rZ) implies that τA(ψ) = 0.

Then:

θQ,ψ | Wr2s(Q) = C1∑ψ(detM)

∑QT ≡ 0 mod s(Q)

T mod r2s(Q)

exp(2πir tr(tM QTs(Q)))

θ(z, r4s(Q)Q : T )

= C1∑

T τ QTs(Q)

(ψ)θ(z, r4s(Q)Q : T )

= C1τ1(ψ)∑

T ψ−1(det QT

s(Q))θ(z, r4s(Q)Q : T )

but:

θ(z, r2Q : T ) =∑

N

exp(πi tr(zt(r2QN +QT

s(Q))s(Q)Q−1(r2QN +

QT

s(Q))))

so that:

θQ,ψ | Wr2s(Q)(z) = C1τ1(ψ)∑ψ−1(detX)

∑N exp(πi tr(zt(r2QN +X)s(Q)Q−1(r2QN +X)))

where X = X(T ) = QTs(Q) and where T runs over elements of Mg×g(Z) modulo r2s(Q) with

QT ≡ 0 mod s(Q).

Now let I be the right ideal in Mg×g(Z) generated by r2Q. Then the sets T mod

r2s(Q) | QT ≡ 0 mod s(Q) and X mod I are in bijection by the maps:

T 7−→ s(Q)−1QT

s(Q)Q−1X ←− X

Thus:

θQ,ψ |Wr2s(Q)(z) = C1τ1(ψ)∑

XmodI ψ−1(detX)

∑N≡XmodI exp(πi tr(z

tNs(Q)Q−1N))

= C1τ1(ψ)θs(Q)Q−1,ψ−1

Now we must return to the two claims about the sums τA(ψ). The first is easy on making

a change of variable. For the second we shall show the existence of B ∈ GLg(Z/rZ) such that

BA = A and ψ(detB) 6= 1. From this one sees easily that τA(ψ) = ψ(detB)τA(ψ) and so

68

τA(ψ) = 0. To find such a B let M be the submodule of (Z/rZ)g generated by the columns

of A. What we require is β ∈ Aut((Z/rZ)g) with β |M= Id and with ψ(det β) 6= 1. However

we can pick a basis e1, ..., eg of (Z/rZ)g such that M = (λ1e1, ..., λgeg) with (λ1, r) 6= 1.

Then let β be represented by :

α 0 . . 0

0 1 0

. . .

. . .

0 0 . . 1

with α ≡ 1 mod r(λ1,r)

and ψ(α) 6= 1. This is the desired β and completes the proof of the

lemma.

We now introduce a formal q-expansion which looks like an Eisenstein series:

Eφ,b,s(k, α) =∑

(det h, p∞) = ps

h ∈ symm∗g(Z)

h > 0

ahqh

where:

• b is prime to p

• α : (Z/ptZ)×〈〉−→ (1 + pZ)/(1 + ptZ) −→ Qac×

• φ(Z/bpZ)× −→ Qac× and φ = φ(p)φ(p) corresponding to (Z/bpZ)× ∼= (Z/bZ)× ×(Z/pZ)×

and where:

ah = (det h)−g+12 f

g2

˜φ(p)ξ

(p)h

φ(p)ξ(f ˜φ(p)ξ

(p)h

)τ(˜φ(p)ξ

(p)h )L(p)(1 +

g2 − k, ˜φ−1αω−kξh)

α−1(f ˜φ(p)ξ

(p)h

)〈f ˜φ(p)ξ

(p)h

〉−kα(p−s det h)〈p−s det h〉k

∏l ∈ P (h)

l 6 |bp

Ml,h(φ(l)α(l−1)〈l〉−k)

∏l 6 |f ˜

φ(p)ξ(p)h

l | b

(1− l g2 φξh(l)α(l−1)〈l〉−k)

69

Here we have used the notation ξh = ξξph where ξph is defined modulo a number prime to p

and where ξ = ωsp−12 . Also L(p) denotes the L-series with the Euler factor at p removed.

Now we prove that these are not too far off from being modular forms.

Lemma 3.8 Let b be prime to p, φ : (Z/bpZ)× → Qac×, φ = φ(p)φ(p) as above with

φ(p) 2 6= 1, φ(−1) = 1. Let α : (Z/pt1Z)×〈〉→ (1 + pZ)/(1 + pt1Z) → Qac× also be as

above. Let Q ∈ symm∗g(Z) with (detQ, p∞) = pt2 . Set t = max(t1, t2 + 1). Also let

ψ : (Z/rZ)× → Qac× be such that s(Q)r2 | bpt and p | r. Finally assume that k ≥ g2 + 1.

Then:

CU tp(θQ,ψEφ,b,t2(k, α)) = U tp(θQ,ψα2ω−2k Ek,φα−1ωk ,bp)

∈ Mk+ g2(Γ0(bp

t), ψφχQαω−k)

with C a non-zero constant.

We first separate out a part of the proof:

Lemma 3.9 Let β be a character defined mod pt1 and let Q ∈ symm∗g(Z) with Q > 0 and

(detQ, p∞) = pt2 . Then if t = max(t1, t2 + 1) and if A =∑Ahq

h is a formal q-expansion:

β(p−t2 det(−Q))U tp(θQ,ψβ2A) = U tp(θQ,ψAβ)

where Aβ =∑

(det h,p∞)=pt2 ψ(p−t2 deth)Ahq

h.

Proof:We look at the coefficient of qh on both sides. On the left hand side we get:

∑

g+tnQn=pth

β2ψ(detn)Agβ(p−t2 det(−Q))

while on the right hand side we get:

∑

g+tnQn=pth

ψ(detn)Agβ(p−t2 det g)

This proves the lemma.

Proof of lemma three:

First note that:

θQ,ψα2ω−2k ∈Mk(Γ0(bpt), ψχQα

2ω−2k)

70

and that as φ(p) 2 6= 1 and k ≥ g2 + 1 we have:

Ek,φα−1ωk ∈Mk(Γ0(bpt), φα−1ωk)

Thus we need only show that the claimed identity holds (formally). However:

αω−k(detQ)U tp(θQ,ψα2ω−2k Ek,φα−1ωk) = U tp(θQ,ψE1)

= ǫ(p−t2 detQ)U tp(θQ,ψE2)

where:

• E1 =∑

(det h, p∞) = pt2

h ∈ symm∗g(Z)

h > 0

bhqh

• E2 =∑

(det h, p∞) = pt2

h ∈ symm∗g(Z)

h > 0

ǫ(p−t2 deth)bhqh

• bh = fg2−k

˜φ(p)ξα−1ωkφ(p)(f ˜φ(p)ξα−1ωk

)τ( ˜φ−1(p)ξαω−k)ah

.

1 f ˜φ(p)ξα−1ωk= p2m

ξ(p)h (p) f ˜φ(p)ξα−1ωk

= p2m−1

Aξ(p)h (p)

ξφ(p)ωkα−1 = 1

with m a positive integer

• A± = 1∓φ(p)(p)pg2−k

1∓φ(p)(p)−1pk−g2−1

• ǫ = ωp−12

Here we have used the fact that if χ1, χ2 are primitive characters with coprime conductors

then τ(χ1χ2) = τ(χ1)τ(χ2)χ1(fχ2)χ2(fχ1). Note that A± 6= ∞ or 0 as φ(p) 2 6= 1 and also

that:

ξ(p)h (p) = δ.ǫ(p−t2 deth)

with δ = ǫ(−1) g2 if p ≡ 1 mod 4 and δ = ǫ(−1) g2+t2 if p ≡ 3 mod 4.

Thus we see that:

71

• If f ˜φ(p)ξα−1ωkis an even power of p and is 6= 1 then:

E1 = fg2−k

˜φ(p)ξα−1ωkφ(p)(f ˜φ(p)ξα−1ωk

)τ( ˜φ(p)ξα−1ωk)Eφ,b,t2(k, α)

• If f ˜φ(p)ξα−1ωkis an odd power of p then:

E2 = δfg2−k

˜φ(p)ξα−1ωkφ(p)(f ˜φ(p)ξα−1ωk

)τ( ˜φ(p)ξα−1ωk)Eφ,b,t2(k, α)

• If ˜φ(p)ξα−1ωk = 1 then:

E1(A−1δ +A−1−δ) + E2(A

−1δ −A−1−δ) =

2fg2−k

˜φ(p)ξα−1ωkφ(p)(f ˜φ(p)ξα−1ωk

)τ( ˜φ(p)ξα−1ωk)Eφ,b,t2(k, α)

In the first two cases we conclude at once the desired equality. In the last case we conclude

that:

A−1ǫ(p−t2 detQ)δ

αω−k(p−t2 detQ)U tp(θQ,ψα2ω−2k Ek,φα−1ωk ,bp) =

fg2−k

˜φ(p)ξα−1ωkφ(p)(f ˜φ(p)ξα−1ωk

)τ( ˜φ(p)ξα−1ωk)Utp(θnQ,ψEφ,b,t2(k, α))

and again we are done.

We shall now fix a cusp form f in Sk(Γ0(q), χ). Assume that f has Fourier expansion∑

h∈symm∗g(Z)h>0 ah(f)q

h. Also if Q ∈ symm∗g(Z), Q > 0 and ψ : (Z/rZ)× → Qac× we

introduce a Dirichlet series:

DQ,ψ(s) = L(s+ g2 , ψχχQ)

∏ g2−1

0 L(2s+ 2i, ψ2χ2)∑

M∈SLg(Z)\M+g×g(Z)

ψ(detM)aMQtM (f)(detM)1−k−s

Lemma 3.10 DQ,ψ(s) =∑∞

n=1dnns with |dn| ≤ Cng+

12 log n for some constant C. In par-

ticular it is convergent for Re s > g + 32 .

Proof:We shall establish the following claims:

1. |ah(f)| ≤ C1(det h)k2

2. #[M ] ∈ SLg(Z)\M+g×g(Z)|detM = n ≤ C2n

g− 12 log n

72

From these two it at once follows that:

∑

M∈SLg(Z)\M+g×g(Z)

aMQtM (f)(detM)1−k−s =∑ an

ns

with |an| ≤ C1C2(detQ)k2ng+

12 log n, and hence that:

|dn| ≤∑

n=n1n2|an1 |

∑n2=n3m2

1...m2g2

1

≤ C1C2(detQ)k2ng+

12 (log n)

∑n=n1n2

n−g2

∑n2=n3m2

1...m2g2

1

≤ C1C2(detQ)k2ng+

12 (log n)

(∑nn3=1 n

−g3

) (∑n1 m

−2g) g

2

≤ Cng+12 log n

We now prove the two claims that we made. For the first recall that:

ah(f) =

∫

X∈symmg(R)/symmg(Z)f(X + iY ) exp(−πi tr(zh))

where z = X + iY and Y is constant. Thus from the bound given on page 335 of [Sm] we

see that:

|ah(f)| ≤ C3(detY )−k2 exp(π tr(Y h))

for any Y ∈ symmg(R) with Y > 0. In particular if h12 denotes the positive definite square

root of h, putting U = h12Y h

12 we see that:

|ah(f)| ≤ C3(det h)k2 (detU)−

k2 exp(π trU)

for any U > 0, as required.

For the second claim, we see that:

#[M ] ∈ SLg(Z)\M+g×g(Z)|detM = n =

∑n=n1...ng

1n2n23...n

g−1g

≤ ng−1∑

n=mng

(∏g−1i=1

∑mj=1 j

−i)

≤ Cng−1(log n)#m |m|n, m > 0≤ 2Cng−1(log n)n

12

Lemma 3.11 Let f , Q, ψ be as above with:

• ψ is primitive modulo r

73

• q|s(Q)r2

• all prime divisors of qs(Q) divide r

• ψχ(−1) = (−1)k

• detQ 6= 0

• either k ≥ g + 1 and (ψχ)2 6= 1, or k ≥ g + 3

then:

DQ,ψ(k − g) = C(f, θs(Q)Q−1,ψ−1Ek− g2,χψχQ,b)b

where C 6= 0 and b = s(Q)r2.

Proof:According to proposition 2.3 of [AK]:

DQ,ψ(s) = c1cs2γ(s)

−1L(s)〈f, θQ,ψE(z,s

2+g

2− k

2; k − g

2, χψχQ, b)〉b

with c1 6= 0 6= c2, γ(s) =∏g

1 Γ(s+k−i

2 ), L(s) = L(s + g2 , ψχQχ)

∏ g2−1

0 L(2s + 2i, χ2ψ2) and

Re s sufficiently large. By proposition 2.4 of [AK] the equation remains valid whenever all

the terms remain defined. The result now follows from lemmas one and two and the facts

that:

• 〈A,B〉 = 〈A|Wb, B|Wb〉

• χψχQ(−1) = (−1)k+ g2

• Eck,χ,b = (−1) g2χ(−1)Ek,χ−1,b (as τ(χ)c = χ(−1)τ(χ−1) ).

Lemma 3.12 Let f be a cusp form in Sk(Γ0(q), χ) with k ≥ 2g+2, which is an eigenvalue

of the Hecke operators and such that ef = f . Let Q ∈ symm∗g(Z), Q > 0 be such that

aQ(f) 6= 0. Let pt2 = (p∞, s(Q)g detQ−1). Choose r such that it is divisible by p2 and by

all prime divisors of s(Q) and such that q|r2s(Q). Write s(Q)r2 = bpt1 with b prime to p.

Let T ∈ Tr with Tf 6= 0.

74

Then we can choose ψ a primitive character modulo r such that ψ(−1) = (−1)kχ(−1)and (χψ)2 6= 1. Write ψχχQ = φωkα−1 with φ a character modulo bp, ω the Teichmuller

character and α : (Z/pt1Z)×〈〉→ (1 + pZ)/(1 + pt1Z)→ Qac×. Then:

(f, Te(θs(Q)Q−1,ψ−1ω2kα−2Eφ,b,t2(k −g

2, α)))bpt1 6= 0

where e(θE) is defined as U−tp eU tp(θE) for t ≥ max(t1, t2 + 1).

Proof:We know from Lemma 3 that the left hand side is a non-zero multiple of:

(f, Te(θs(Q)Q−1,ψ−1Ek− g2,χψχQ,bp))bpt1

(Tef, θs(Q)Q−1,ψ−1Ek− g2,χψχQ,bp)bpt1

which is itself a non-zero multiple of DQ,ψ(k − g). However we know from [A1] that in this

case DQ,ψ(s) has an Euler expansion of degree 2g + 1. The result thus follows from the

following fact:

Let D(s) =∑ dn

ns be a Dirichlet series with |dn| ≤ Cna. Assume that for Re s

sufficiently large D(s) =∏pQp(p

−s)−1 with Qp a polynomial of bounded degree.

Then D(s) is non-zero and convergent for Re s > a+ 1.

3.4 The General Strategy

This method of constructing Hida families is due to Wiles (see [Wi]).

Fix a positive integer N and a character χ : (Z/NpZ)× → Qac×. Let O denote the

integers in a finite extension of Qp containing all the ϕ(N) roots of unity (here ϕ is Euler’s

phi-function), and set Λ = O[[T ]]. Also let Υ denote the set of pairs (k, α) where k is an

integer ≥ g + 1 and α : (1 + pZp) → Qac× is a character of finite order, so that we may

think of α as a character:

(Z/prZ)×〈〉−→ (1 + pZ)/(1 + prZ) −→ Qac×

We shall denote the smallest possible choice of r by r(α).

75

Now for I an infinite subset of Υ set:

MI(N,χ,Λ) =

F ∈ Λ[[q]]g | F |T=(α(1+p)(1+p)k−1) ∈Mk(Γ0(Npr(α)), χω−kα,Oα) for (k, α) ∈ I

where Oα denotes the integers of the field FO(α(1 + p)). Also set:

M(N,χ,Λ) =

F ∈ Λ[[q]]g | F |T=(α(1+p)(1+p)k−1) ∈Mk(Γ0(Npr(α)), χω−kα,Oα)

for all but finitely many (k, α) ∈ Υ

i.e.

M(N,χ) =⋃

I of finite complement inΥ

MI(N,χ)

Note that:

• λ ∈ Λ, F ∈ Λ[[q]]g and λF ∈ M implies that F ∈ M

• if F ∈ M then there is a λ ∈ Λ such that λF ∈ MΥ.

• MI/(1+T −α(1+p)(1+p)k)M →Mk(Γ0(Npr(α)), χω−kα,Oα) if Υ− I is finite and

(k, α) ∈ I.

Before giving examples of elements of M(N,χ), or as we shall write Λ-adic forms, we

must introduce Hida’s idempotent in this context. First we define an action of Up on Mcompatible with specialisation by setting:

Up(∑

h

Ahqh) =

∑

h

Aphqh

Then we have:

Lemma 3.13 Let I ⊂ Υ be infinite. Then

1. there is a unique operator e onMI compatible with Hida’s idempotent under special-

isation

2. eF = limr→∞Ur!p F

76

3. Up is invertible on eMI .

Proof:We have the specialisation map:

φ :MI →⊕

(k,α)∈I

Mk(Γ0(Npr(α)), χω−kα,Oα)

The image is certainly preserved by Up. First we shall show that the image is also closed.

For this let Fj ∈ MI , φ(Fj)→∏I gi as j →∞. Then if θ : Λ→∏

I Oα is the specialisation

map we have that θ(ah(Fj)) →∏ah(gi) as j → ∞. However Λ is compact and so θΛ is

closed in∏I Oα. Thus we can find bh ∈ Λ such that θbh =

∏ah(gi). Then it is easily seen

that∑bhq

h ∈ MI and that φ(∑bhq

h) =∏I gi.

Thus the operator e = limr→∞ Ur!p preserves φMI and this proves the first two parts. It

is easily seen that Up is injective on eMI . Finally if F ∈ eMI then forG = limr→∞ Ur!−1p F ∈

MI , which exists as limr→∞Ur!−1p φF exists, UpG = eF = F . This proves the last part also.

Corollary 3.1 The Hida operator defined in the lemma satisfies the following properties:

1. e2 = e

2. if t commutes with Up then it also commutes with e

3. if I ⊃ J the action of e is compatible with MJ ⊃MI

4. e extends to an operator on M with the same properties.

These are all easy.

We are now in a position to construct some examples of Λ-adic forms:

Example 3.1 Let

• b be prime to p

• φ : (Z/bpZ)× → Qac× be a character with:

– φ(−1) = 1

– φ = φ(p)φ(p) its decomposition into “at p” and “ away from p” parts

77

– φ(p) 2 6= 1

• Q ∈ symm∗g(Z), Q > 0

• (detQ, p∞) = pt2

• N and r be such that p|r, s(Q)r2|N and N = bpt3

Then there is a Λ-adic form GN,Q,ψ,φ ∈ MΥ(b, ψφχQ) such that:

GN,Q,ψ,φ|T=(α(1+p)(1+p)k−1) = e(θQ,ψEφ,b,t2(k −g

2, α))

in the notation of section two.

Proof:For t ≥ max(t3, t2 + 1) set It = (k, α) ∈ Υ | r(ǫ) ≤ t. Then we can define

G(t) ∈ MIt to be U tp(θQ,ψ∑

(det h, p∞) = pt2

h ∈ symm∗g(Z)

h > 0

Bh(T )qh) where:

Bh(T ) = (det h)−(g+12)f g

˜φ(p)ξ

(p)h

φ(p)ξ(f ˜φ(p)ξ

(p)h

)τ(˜φ(p)ξ

(p)h )

(1 + T )

logp(f−1

˜φ(p)ξ

(p)h

p−t2 det h)

Gφ−1ξhω

−g2( (1+T )(1+p)g − 1)

∏l ∈ P (h)

l 6 |bp

Ml,h(φ(l)〈l〉g2 (1 + T )− logp l)

∏l 6 |f ˜

φ(p)ξ(p)h

l|b

(1− l g2 〈l〉 g2 φξh(l)(1 + T )− logp l)

and where:

• x = ω(x)(1 + p)logp x for x ∈ Z×p

• Gχ(α(1 + p)(1 + p)n − 1) = L(p)(1 − n, χαω−n) for α as above, n > 0 an integer and

χ(−1) = 1.

That Gt is in fact inMIt follows from lemma 3.8 of section 3.3. Then U−tp eG(t) ∈ MIt

and U−tp eG(t)|T=(α(1+p)(1+p)k−1) = e(θQ,ψEφ,b,t2(k − g2 , α)) again by lemma 3.8 of section

78

3.3. In particular if t′ > t then:

U−tp eG(t) ∈ MIt

‖ ∪U−t

′

p eG(t′) ∈ MIt′

and so G = U−tp eG(t) ∈ ⋂t′≥tMIt′ =MΥ and has the desired specialisations.

Next we record two technical results about Λ-adic forms:

Lemma 3.14 1. eM(Np,χ) = eM(N,χ)

2. If O′ denotes the integers of a finite extension of O and Λ′ denotes O′[[T ]] then:

M(N,χ,Λ) ⊗Λ Λ′ =M(N,χ,Λ′)

These are both straightforward. We can use part two to defineM(N,χ,R) for any Λ-algebra

R to beM(N,χ,Λ) ⊗Λ R.We now prove one of the main theorems on Λ-adic forms:

Theorem 3.1 M(N,χ) is a finite free Λ-module.

Proof:We may and shall assume that N is prime to p (by the last lemma). Note that

M(N,χ) is certainly torsion free over Λ.

Step 1 There is a constant C such that dim eMk(Γ1(Np)) ≤ C for all k.

Let:

Γ(g) =

A B

C D

∈ Sp2g(Z) |C ≡ 0 mod Np, B ≡ 0 mod N,

A ≡ D ≡ modN, detA ≡ 1 mod p

Then it will do to show that dim eMk(Γ(g)) is bounded independently of k. However we know

from proposition 2.1 that for each g′ there is a constant Cg′ such that dim eSk(Γ(g′)) < Cg′ .

The result now follows by induction using lemma 3.5 of section 3.2.

Step 2 If C is as in step 1 then M(N,χ) does not contain more than C Λ-linearly

independent elements.

79

Assume that F0, ..., FC were linearly independent elements of M(N,χ). Then we can

find h0, ..., hC in symm∗g(Z) such that the matrix (ahi(Fj)) is non-singular. Then for k

sufficiently large:

• Fj |T=((1+p)k−1) ∈ eMk(Γ1(Np),O)

• det(ahi(Fj |T=((1+p)k−1))) 6= 0

For such k the Fj |T=((1+p)k−1) are (C+1) linearly independent elements of eMk(Γ1(Np),O), which is a contradiction.

Step 3M(N,χ) is a compact finitely generated Λ-module.

We can choose F1, ..., Fr a maximal set of linearly independent elements of M(N,χ),and we can choose h1, ..., hr in symm∗g(Z) with λ = det(ahi(Fj)) 6= 0. Then we claim that

λM(N,χ) ⊂ 〈F1, ..., Fr〉Λ. For if F =∑bjFj with bj ∈ FΛ then we have the non-singular

set of equations:

ahi(F ) =∑

bjahi(Fj)

for “unknowns” bj. As the ahi(F ) and the ahi(Fj) are in Λ, bj ∈ λ−1Λ.Step 4 In particularM(N,χ) =MI(N,χ) for some subset I of Υ with finite comple-

ment. Thus for almost all pairs (k, α) ∈ Υ we have that:

M(N,χ)/(T − α(1 + p)(1 + p)k)M(N,χ) →Mk(Γ1(Npr(α)),Oα)

Thus the theorem follows from the following lemma:

Lemma 3.15 Let M be a compact Λ-module and ℘i an infinite collection of height one

primes such that M/℘iM is a finite torsion free Zp-module then M is a finite free Λ-module.

Proof:Let Oi = Λ/℘i. Then M/℘iM is a finite free Oi-module. Let M/℘iM ∼= Orii . Let

r = ri0 = min ri. Then by Nakayama’s Lemma Λrβ→→ M , and Ori →→ (M/℘iM) for all i.

Thus r = ri for all i and Ori∼→ (M/℘iM). Thus if ~λ ∈ Λr is such that β(~λ) = 0 then ~λ ∈ ℘ri

for all i and so ~λ = ~0. That is β is an isomorphism.

80

Corollary 3.2 There exists λ ∈ Λ such that:

M(N,χ) ⊃MΥ(N,χ) ⊃ λM(N,χ)

Proof:If F ∈ M(N,χ) then we know that we can find λ ∈ Λ so that λF ∈ MΥ(N,χ), this

is enough asM(N,χ) is finitely generated over Λ.

We shall now define an action of the Hecke ring TNp on M(n, χ) (or MI(N,χ)) by

setting:

F |T (n) = n−g(g+1)

2 χ(ng)(1 + T )g logp n∑

d1|d2|...|dg|ndg1...dg

∑D χ(detD)−1(1 + T )− logp detDan−1DhtD

where the second sum is taken over a set of representatives for:

GLg(Z)\GLg(Z)

d1. . .

dg

GLg(Z)

This is compatible with specialisation. We can similarly define an action of T∗Np on

M(N,χ) ⊗O Qacp which is compatible with specialisation, by using the formula in section

3.2.

Lemma 3.16 TNp acts semi-simply on M(N,χ, FΛ).

Proof:It will do to show that each T (n) ∈ TNp acts semi-simply, because TNp is commuta-

tive. Let T (n) have characteristic polynomial P (X), and let Q(X) ∈ Λ[X] be the product

of the distinct irreducible factors P (X). It will do to show that Q(T (n)) acts as 0 on Mor equivalently on M/(1 + T − α(1 + p)(1 + p)k)M for infinitely many (k, α). However

for almost all (k, α) ∈ Υ, M/(1 + T − α(1 + p)(1 + p)k)M → M. For such a pair (k, α)

let P (X) and Q(X) ∈ Oα[X] denote the reduction of P and Q. Let q be defined from P

the same way Q was defined from P . Then q(X)|Q(X). Moreover q(T (n)) = 0 as P (X) is

the characteristic polynomial of T (n) on a subspace of Mk(Np?, χαω−k) on which T (n) is

known to act semi-simply. Thus Q(T (n)) = 0 and we are done.

Before proving our second main theorem about lifting eigenforms we state and prove an

algebraic lemma which we shall require:

81

Lemma 3.17 Let R denote the integers of a finite extension of Qp. LetM be a finite torsion

free R module. Let T be a commutative ring of operators on M . Let M⊗FR =⊕

I Vi where

T acts on Vi by a character λi : T → R. Let a : M → R be a linear form, x ∈ M and

λ : T → R be such that:

• (λ(t)− t)x ∈ pAM

• valp a(x) ≤ B

• rkM ≤ C

Then for some i:

• valp (λ(t)− λi(t)) ≥ A−BC for all t ∈ T

• a 6≡ 0 on Vi

Proof:Let J = i ∈ I | a|Vi 6≡ 0. Let π denote the projection of M to⊕

J Vi. Then

• (λ(t)− t)π(x) ∈ pAπM

• valp a(π(x)) ≤ B

• rkπM ≤ C

Thus we may assume that a 6≡ 0 on all Vi. Let M′ =

⊕M ∩ Vi. Let I T and I ′ R be

the annihilators of x in M/pAM . Then T/I ∼= R/I ′ via λ. Then we have that:

FittT (M′) = 0

and so FittT/I(M′/IM ′) = 0

and so FittR/I′(⊕

(M ∩ Vi)/λi(I)(M ∩ Vi)) = 0

and so∏λi(I)dimVi ⊂ I ′ ⊂ pA−B

and so for some i, valp λi(I) ≥ A−BC

as desired. (Here Fitt denotes the Fitting ideal. For some of its properties see (for example)

the appendix of [MW1].)

82

Theorem 3.2 Let f ∈ eSk(Nps, χαω−k,O) be an eigenform of the Hecke algebra TNp, say

f |T (n) = λ(n)f . Let Q ∈ symm∗g(Z), Q > 0 be chosen such that aQ(f) 6= 0. Let M be

chosen prime to p such that N |M and such that if l (6= p) is any prime dividing s(Q), say

lβ||s(Q), then lβ+2i||M with i some positive integer. Then there exists a finite extension

of FΛ, with integers R say, and a prime ℘ of R above (1 + T − α(1 + p)(1 + p)k) and

F ∈ M(M,χ,R) such that:

• F |T (n) = λ(n)F with λ(n) ∈ R

• λ(n) ≡ λ(n) mod ℘

for all n such that T (n) ∈ TMp.

Proof:We shall first show the result for sufficiently large k and then deduce it for all k.

Choose k0 such that:

• k0 ≥ 2g + 2

• M =MI for some I containing (k, α) for all k ≥ k0 and for all α.

Step 1 If k ≥ k0 then we can find a non-zero f ′ = M(M,χ)/(1 + T − α(1 + p)(1 +

p)k)M(M,χ) with f ′|T (n) = λ(n)f ′ for all T (n) ∈ TMp.

We may assume f is an eigenvector for T∗Mp. Then we can find T ∈ TMp with f |T =

µf 6= 0 and such that TMk(Mpt, χαω−k) is an eigenspace for TMp. Then choose r and t

such that (Nps)|s(Q)r2 = Mpt1 and p2|r and choose ψ a primitive character modr with

ψ(−1) = χ(−1) and (χ(p)ψ(p))2 6= 1. Set φ = ψα2ω−2kχQ. Then we see from example 3.1

and lemma 3.12 and section 3.3 that:

0 6= (f, TeGMpt1 , s(Q)Q−1, ψ−1ω2kα−2, φ|T=(α(1+p)(1+p)k−1))Mpt1

Thus f ′ = TeGMpt1 , s(Q)Q−1, ψ−1ω2kα−2, φ|T=(α(1+p)(1+p)k−1) is non-zero and will do.

Step 2 If k ≥ k0 then the theorem is true.

Choose R the integers in a finite extension of FΛ such that TMp is diagonalisable on

M(M,χ,FR). Let e1, ..., er ∈ M(M,χ,R) be eigenvectors of TMp which span

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M(M,χ,FR). Let λi : TMp → R be the eigenvalue corresponding to ei. Let U =⊕Rei

and let T ⊂ EndR(M(M,χ,R)) be the R-subalgebra generated by TMp. Then U is a

faithful T module. Let I T be the annihilator of f ′. Then arguing as in the proof

of lemma 3.17 we see that there is a prime ℘ above (1 + T − α(1 + p)(1 + p)k) such that

FittR/℘(U/IU) = 0. Then∏

λi(I) ⊂ ℘ and so for some i λi(I) ⊂ ℘, i.e. for all T (n) ∈ TMp

λi(T (n)) ≡ λ(T (n)) mod ℘.

Step 3 The theorem is true for all k.

We have seen (proposition 2.1) that there is a constant C such that

dimMl (Npt, χαω−k, FO) < C for all l. Let valp (aQ(f)) = B. Let θ be the theta se-

ries defined in lemma 2.3. Then θprf ∈ Mk+(p−1)pr(Np

t, χαω−k,O). Moreover θprf ≡

f mod pr+1, and so e(θprf) ≡ f mod pr+1 and:

(θprf)|k+(p−1)prT (n) ≡ f |kT (n)

≡ µ(n)f

≡ µ(n)(θprf) mod pr+1

as ak+(p−1)pr ≡ ak mod pr+1 for all a. Thus also ((θprf)|e)|T (n) ≡ µ(n)((θprf)|e) mod pr+1.

Thus by the lemma proved just before this theorem we see that for r > B we can find

fr ∈Mk+(p−1)pr(Npt, χαω−k,O) such that:

• fr|T (n) = λr(n)fr

• λr(n) ≡ λ(n) mod p[r+1−BC ]

• aQ(fr) 6= 0

Now for r sufficiently large this implies that we can find Fr ∈ M(M,χ,R) (whereR denotes

the integers in an extension of Λ such that TMp is diagonalisable on M(M,χ,FR)) such

that:

• Fr|T (n) ≡ µr(n)Fr

• µr(n) ≡ λ(n) mod (℘r, p[ r+1−B

C ])

• ℘r is a prime of R above (1 + T − α(1 + p)(1 + p)k+(p−1)pr).

84

Now there are only finitely many choices for µr and so there is an infinite set S of

positive integers such that for r ∈ S µr = µ. Then:

µ(n) ≡ λ(n) mod⋂

S

(℘r, p[ r+1−B

C ])

We claim that:⋂

S

(℘r, p[ r+1−B

C ]) ∩ Λ ⊂ (1 + T − α(1 + p)(1 + p)k)

from which the theorem would follow. However this inclusion follows from the two facts:

• (℘r, p[ r+1−B

C ]) ∩ Λ = (p[r+1−BC ], (1 + T − α(1 + p)(1 + p)k+(p−1)pr))

• (p[r+1−BC ], (1+ T −α(1 + p)(1+ p)k+(p−1)pr)) ⊂ (p[

r+1−BC ], (1 + T −α(1+ p)(1+ p)k))

The first of these is easy and the second not much harder. (In general if R1 ⊂ R2 are two

rings with prime ideals ℘1 and ℘2 where ℘1 = ℘c2, and if a ∈ R1 then (a, ℘2)c = (a, ℘1). To

prove this one reduces at once to the case R1, R2 integral domains and ℘1 = ℘2 = 0, in

which case it is obvious.)

3.5 Conjectural Applications

Throughout the rest of this paper we shall restrict to the case g = 2, i.e. to GSp4. This is just

for definiteness and because g = 2 was the case of interest for us. For other even g exactly

similar results should hold with ρ : Gal(Qac/Q) → GSp4 replaced with ρ : Gal(Qac/Q) →GLN for suitable N , and with a suitable characteristic polynomial. We shall also make free

use of the following conjecture:

Conjecture 3.1 Let f ∈Mk(N,χ) be an eigenform of the Hecke algebra TN , say f |T (n) =λ(n)f . Assume k ≥ 3. Then there is a continuous semi-simple representation:

ρ : Gal(Qac/Q) −→ GSp4(Qacp )

unramified outside pN and such that the characteristic polynomial of Frobl for l 6 |Np is

given by:

X4 − λ(l)X3 + (λ(l)2 − λ(l2)− l2(k−2)χ(l2))X2 − l2k−3χ(l2)λ(l)X + l4k−6χ(l4)

85

Our aim in this section is to show that these conjectures would imply similar results for

ordinary Λ-adic forms and for ordinary eigenforms of weight two. We first prove a general

result on lifting group representations:

Proposition 3.1 Let R be an integral domain, ℘ii∈I be an infinite set of prime ideals

such that the intersection of any infinite subset is zero. Let Oi denote a finite extension of

R/℘i containing its integral closure, and assume that FOi is of characteristic zero. Let Γ be

a group and ρi : Γ→ GLN (Oi) a semi-simple representation. For each x ∈ Γ suppose there

exists Tx ∈ R with Tx ≡ tr(ρix) mod ℘i.

Then there is an infinite subset J ⊂ I, S an integral domain finite over R[f−1] for some

f ∈ R, a semi-simple representation ρ : Γ → GLN (S) and for each i ∈ J a prime Pi over℘i, a field Li ⊃ FOi , a map S/Pi → Li such that:

S/Pi → Li⋃ ⋃

R/℘i ⊂ Oi

commutes and such that ρ mod Pi is conjugate to ρi as a representation into GLN (Li).

We break up the proof into a series of lemmas.

Lemma 3.18 Let K be a field of characteristic zero, let A ⊂ MN (K) be a split semisim-

ple sub-algebra and let H ⊂ GLN (K) be its normaliser. Let a1, ..., ar and b1, ..., brbe subsets of A each of which spans A and assume that for s = 1, ..., 4 we have that

tr(∏sj=1 aij ) = tr(

∏sj=1 bij) for all s-tuples (i1, ..., is) ∈ 1, ..., rs. Then there is an el-

ement n ∈ H with nain−1 = bi for all i.

Proof:A is of the form⊕

αMcα(K)aα and without loss of generality we may assume

(mαβ)α; β=1,...,aα embeds as⊕m⊕bααβ , where

∑α aαbαcα = N . Also assume that the notation

is such that for α 6= α′ (bα, cα) 6= (bα′ , cα′).Let eαβ denote the idempotents corresponding to

our decomposition of A into simple algebras. Let eαβ =∑λαβiai and set fαβ =

∑λαβibi

Then we see that:

86

• e2αβ = eαβ ⇒ tr(e2αβ − eαβ)aj = 0 ∀j⇒ tr(f2αβ − fαβ)bj = 0 ∀j⇒ f2αβ = fαβ

• eαβ central ⇒ tr(eαβaiaj) = tr(aieαβaj) ∀i, j⇒ tr(fαβbibj) = tr(bifαβbj) ∀i, j⇒ fαβ central

• cα = rk (tr(eαβaiaj))i,j ⇒ cαrk (tr(fαβbibj))i,j

⇒ cα = dim fαβA

• treαβ = cαbα ⇒ trfαβ = cαbα

From this we see that for each α fαβ is a permutation of eαβ. We can conjugate

the bi’s by an element of H such that for each α, β fαβ = eαβ. Thus we may without

loss of generality assume that this equation holds. Now fix α, β and set a′i = eαβai and

b′i = eαβbi. Then we can consider a′i and b′i ∈ Mcα(K) such that for s = 1, 2 or 3 and

(i1, ..., is) ∈ 1, ..., rs tr(∏sj=1 a

′ij) = tr(

∏sj=1 b

′ij). Moreover we need only show that we

can find n′ ∈ GLcα(K) with n′a′in′−1 = b′i for all i.

Let ǫkl =∑λklia

′i and set δkl =

∑λklib

′i. Then as above we can show that δkk form a set

of commuting idempotents each of trace one and such that∑δkk = 1. Thus by conjugation

by some element of GLcα we may assume that without loss of generality ǫkk = δkk. Then

we also see that ǫkkδkl = δkl and δklǫll = δkl so that δkl = µklǫkl for some µkl ∈ K. Then

µklµlm = µkm and µkk = 1 so there exist νk ∈ K× such that µkl = νkν−1l . Thus we can find

a diagonal matrix n′ in GLcα such that after conjugation by n′ we may assume ǫkl = δkl for

all k and l. Then it is easily seen that a′i = b′i for all i.

Lemma 3.19 Let the assumptions be as in the theorem, but assume further that Γ is gen-

erated by x1, ..., xr and that for each i if Ai is the subalgebra of GLN (Ki) generated by Imρi

then Ai is spanned by ρi(xj). Then the conclusions of the proposition hold, except that

we do not yet claim that ρ is semi-simple.

87

Proof:Without loss of generality we may assume that Γ is the free group on x1, ..., xr .

Ai is semi-simple as it has a faithful semi-simple module. We may assume that Ai is split

over FOi . Then after conjugation and discarding some i we may assume that for all i

Ai = A⊗ FOi where A ∼=⊕Mcα(Z)

aα and a →MN (Z) by:

(xαβ) 7−→ ⊕xbααβ

where the pairs (bα, cα) are distinct for distinct α and where∑aαbαcα = N .

Let Ω = (ai) ∈ Ar | the ai spanA, so that Ω is a subvariety of r∑aαc

2α dimensional

affine space defined over Q. Then by the last lemma we have a map:

Ω −→ AM

given by (ai) 7→ (tr∏sj=1 aij) taken over s = 1, ..., 4 and all (i1, ..., is) ∈ 1, ..., rs. Consider

the point T ∈ AM(R) defined by:

T = (T∏ xij)

Then T mod ℘i is in the image of Ω(FOi) for all i. Thus T is in the image of Ω(F acR ),

because the image of Ω is defined by some polynomial equalities and inequalities. Thus we

can find X1, ...,Xr in A⊗F acR which span A⊗F acR and which satisfy tr∏s

1Xij = T∏ xijfor

all (i1, ..., is) as above. In fact all the Xi lie in some S/R as described in the proposition.

Discard the finite number of ℘i which contain f and assume (as we may ) that FR/FS is

Galois. Choose Pi over ℘i and choose Li such that S/Pi andKi both embed in Li over R/℘i.

Then (Xj mod Pi) ∈ Ω(S/Pi) ⊂ Ω(Li) and φ(Xj mod Pi) = (T∏ xijmod ℘i). Thus by the

last lemma there is n in the normaliser of A in GLN (Li) with n(Xj mod Pi)n−1 = ρi(xj)

for all j and so we are done.

Proof of proposition 3.1

Let A be defined as in the proof of the last lemma. Pick a distinguished index, say 0.

Let x1, ..., xr ∈ Γ be such that ρ0(xj) span A⊗FO0 . Then rk (Txjxk) ≥ dimA and so with

a finite number of exceptions which we may discard rk trρi(xj)ρi(xj) ≥ dimA. However if

aj is a finite subset of A ⊗ F then rk tr(aiaj) ≤ dimA with equality if and only if they

span A. Thus we see that we may assume that the ρi(xj) span A⊗ FOi for all i ∈ I.

88

Now let ∆ be the subgroup of Γ generated by the xi. Then we by the last lemma that

we have a map ρ : ∆ → (A ⊗ S)× ⊂ GLN (S) with the notation as in the proposition and

ρi|∆ ≡ ρ mod Pi. Now define:

A(S′)φ−→ S′

r

for S′ any S algebra by:

a 7−→ (tr aρ(xi))

This is a linear map and the image Ω′(S′) is thus defined by the vanishing of certain linear

forms with coefficients in FS . By inverting some element of S we may assume that φ is

injective and that Ω′ is defined over S. Then for γ ∈ Γ let tγ = (Tγxi). By reduction mod ℘i

for infinitely i we see that tγ ∈ Ω′(S) and hence there exists a unique ρ(γ) with φρ(γ) = tγ .

Then ρ(γ) ≡ ρi(γ) mod Pi for all i ∈ I. Thus we have ρ : Γ→ A⊗ S, ρ ≡ ρi mod Pi for alli ∈ I. It follows at once that ρ is a representation.

Finally we replace ρ by ρss, then at almost all ℘i:

(ρss mod ℘i) = (ρ mod ℘i)ss = ρi

Corollary 3.3 Let Λ be as in section 3.4. Let ℘i be an infinite set of distinct height one

primes. For each i let:

ρi : Gal(Qac/Q) −→ GLN (Qacp )

be continuous representations, unramified outside some integer N . For l 6 |N let cl(X) ∈ Λ[X]

be monic of degree N and such that:

cl(X) ≡ charρi(Frobl)(X) mod ℘i

where chara(X) denotes the characteristic polynomial of a. Then there exists R the integers

of a finite extension of FΛ and:

ρ : Gal(Qac/Q) −→ GLN (FR)

such that c(X) = charFrobl(X) for all l 6 |N .

89

Proof:We first show that the partial map from Gal(L/Q) where L denotes the maximal

extension of Q unramified outside N :

Frobl 7−→ cl(X)

is continuous. Pick any positive integer n and pick some l 6 |N . Then we would like to

show that for l′ with Frobl′ sufficiently close to Frobl we have that cl ≡ cl′ mod mn where

m is the maximal ideal of Λ. But⋂I,Z≥0

(℘i, ps) = 0 and so by compactness there exist

℘1, ..., ℘r and s1, ..., sr such that⋂r

1(℘i, ps) ⊂ mn. Then choose Ui open neighbourhoods

of Frobl such that for l′ with Frobl′ ∈ Ui ρi(Frobl) ≡ ρi(Frobl′) mod psi . Then for l′ with

Frobl′ ∈ U =⋂Ui cl ≡ cl′ mod mn as desired.

Thus we can extend c uniquely to a continuous map:

Gal(Qac/Q) −→ Gal(L/Q) −→ Λ[X]

Then cσ ≡ charρi(σ) mod ℘i for all σ ∈ Gal(Qac/Q) by continuity. Now apply the proposi-

tion and we find a representation ρ : Gal(Qac/Q)→ GLN (FR) for some appropriate R with

ρ mod Pi conjugate to ρi for infinitely many i. Thus charρ(Frobl) ≡ charρi(Frobl) ≡ cl(X) for

infinitely many height one primes, so ρ is the desired representation.

Before proving our last two main theorems we prove one further lemma.

Lemma 3.20 Let F be a field, Γ a group, ρ : Γ → GSp4(F ) ⊂ GL4(F ). Then ρss :

Γ → GL4(F ) preserves a non-degenerate symplectic form, so we may consider ρss : Γ →GSp4(F ).

Proof:Let 0 6= V ⊂ F 4 be a simple Γ submodule. Let 〈〉 denote the symplectic form.

Then we are in one of the following cases:

• V = F 4 and ρss = ρ

• dimV = 2, F 4 = V ⊕ V ⊥. In this case we can choose a basis of F 4 with respect to

90

which 〈〉 is represented by

0 12

−12 0

and ρ is either of the form:

σ 7−→

aσ 0 bσ 0

0 a′σ 0 b′σ

cσ 0 dσ

0 c′σ 0 d′σ

in which case ρss = ρ or it is of the form:

σ 7−→

aσ 0 bσ ∗∗ eσ ∗ ∗cσ 0 dσ ∗0 0 0 fσ

when ρss is easily seen to preserve 〈〉.

• dimV = 2, V ⊥ = V . In this case we can choose a basis of F 4 such that 〈〉 is

given by

0 12

−12 0

and for all σ ρ(σ) =

Aσ Bσ

0 Dσ

. Then ρss is given by

σ 7→

Aσ 0

0 Dσ

and also preserves 〈〉.

• dimV = 1, V ⊥ ⊃ V , dimV ⊥ = 3. In this case we can choose a basis of F 4 with

respect to which 〈〉 is represented by

0 12

−12 0

and ρ has the form:

σ 7−→

eσ ∗ ∗ ∗0 aσ ∗ bσ

0 0 fσ 0

0 cσ ∗ dσ

Then ρss also preserves 〈〉.

91

Theorem 3.3 Assume the conjecture 3.1. Let F ∈ M(N,χ,R), with N prime to p and

R the integers of some finite extension of FΛ, be an eigenform of the Hecke algebra TN , say

F |T (n) = λ(n)F . Then there is a finite extension L of FR and a continuous representation:

ρ : Gal(Qac/Q) −→ GSp4(L)

which is unramified outside Np and such that for l 6 |Np, a prime, Frobl has characteristic

polynomial:

X4 − λX3 + (λ(l)2 − λ(l2)− l−1ν(l))X2 − ν(l)λ(l)X + ν(l)2

where ν(l) = l−3χ(l2)(1 + T )2 logp l.

Proof:Combining conjecture 3.1 and the corollary to the last proposition we at once

deduce the existence of such a representation into GL4(L). We need only show that it

preserves a non-degenerate symplectic form. Let G denote the Zariski closure of Imρ, and

G the connected component of the identity. Let G =∐ρ(γi)G

and let Σ = γ ∈ Γ | ρ(γ) ∈G and trρ(γ) 6= 0. Then G is the Zariski closure of Σ. Now what we require is a matrix

A ∈M4(L) and ǫ : 1, ..., r → ±1 such that:

1. (trρ(γ))ρ(γ)Atρ(γ) = (trρ(γ−1)A for all γ in Σ

2. ρ(γi)Atρ(γi) = ǫi

√det ρ(γi)A

3. tA = −A

4. detA 6= 0

where√det ρ(γi) is some fixed square root of det ρ(γi) which we may assume lies in L. It

is easy to check that a solution to these equations in S/Pi for infinitely many height one

primes Pi of some finitely generated extension of the integers of L contained in L implies

the existence of such a solution in S.( More precisely let rǫ be the dimension of the space of matrices A satisfying con-

ditions 1), 2) and 3). Let e(ǫ)1, ..., e(ǫ)rǫ be a basis of the space of such matrices. That

92

det(∑e(ǫ)iλi) 6≡ 0 may be expressed as Frǫ(e(ǫ)i)(X) 6= 0 for some polynomial Frǫ(e(ǫ)i(X).

Thus we have a solution if and only if rǫ > 0 and Frǫ(e(ǫ)i)(X) 6= 0. Then for some ǫ there

exists a solution of 1)-4) for infinitely many height one primes Pi. This implies rǫ > 0 and

Frǫ(e(ǫ))(X) 6= 0.)

Theorem 3.4 Assume conjecture 3.1. Let f ∈M2 (Npr, χ,O) be an eigenform of the Hecke

algebra TNp, say f |T (n) = λ(n)f . (O the integers in a finite extension of Qp.) Then there

is a multiple M of N and a continuous representation:

ρ : Gal(Qac/Q) −→ GSp4(Qacp )

which is unramified outside Mp and such that if l 6 |Mp then Frobl has characteristic poly-

nomial:

X4 − λ(l)X3 + (λ(l)2 − λ(l2)− χ(l2))X2 − lχ(l2)λ(l) + l2χ(l4)

Proof:By theorem 3.2 we can find an M as above; characters ψ and α such that ψ

is defined modulo Mp, α is of p power order and is defined modulo a power of p and

χ = ψω−2α; R the integers of a finite extension of FΛ; and F ∈ M(M,ψ,R) an eigenform

for the ring of Hecke operators TMp, say F |T (n) = λ(n)F , such that λ ≡ λ(n) mod P with

P a prime of R above (1 + T − α(1 + p)(1 + p)2)). Then by the last theorem we can find a

continuous representation:

ρ : Gal(Qac/Q) −→ GSp4(L)

with L a finite extension of FΛ. Moreover it is unramified outside Mp and for l 6 |Mp the

characteristic polynomial of Frobl is congruent modulo a prime (P ′ say) above (1+T−α(1+p)(1 + p)2) to the polynomial described in the statement of the theorem. As Gal(Qac/Q)

is compact we can find a finite R′ (the integers of L) L ⊂ L4 such that L ⊗R′ L = L4 and

which is preserved by the Galois action. Then LP ′ is free over R′P ′ and we can choose a

93

basis with respect to which the symplectic form is given by:

0 0 µ1 0

0 0 0 µ2

−µ1 0 0 0

0 −µ2 0 0

where µ1|µ2. We may assume that√µ1 and

√µ2 lie in R′, and that µ1 = 1. Let e1, e2, e

′1, e′2

be the corresponding basis. We claim that L′ = 〈e1, µ− 1

22 e2, e

′1, µ− 1

22 e′2〉 is also preserved by

Gal(Qac/Q).

Let (aij) denote an element of Im ρ with respect to the basis e1, e2, e′1, e′2. Then:

(aij)

0 0 1 0

0 0 0 µ2

1 0 0 0

0 µ2 0 0

(aji) = ν

0 0 1 0

0 0 0 µ2

1 0 0 0

0 µ2 0 0

with ν a unit in R′P ′ . From this we see that:

• a11a33 − a13a31 = ν

• a11a23 − a13a21 ≡ 0 mod µ2

• a11a43 − a13a41 ≡ 0 mod µ2

• a21a33 − a23a31 ≡ 0 mod µ2

• a21a43 − a23a41 ≡ 0 mod µ2

• a31a43 − a23a41 ≡ 0 mod µ2

and so:

a23ν = a33(a11a23 − a13a21) + a13(a33a21 − a23a31) ≡ 0 mod µ2

so that a23 ≡ 0 mod µ2. Similarly a21 ≡ a43 ≡ a41 ≡ 0 mod µ2. Thus (aij) also preserves

the lattice L′ as we wanted to show.

94

Now reduction gives us:

ρ : Gal(Qac/Q) −→ Aut(L′ ⊗R′P ′/P ′, 〈〉) ⊂ GSp4(Qacp )

a continuous representation unramified outside Mp with the characteristic polynomials of

the desired forms.

95

Chapter 4

Modular Forms over an Imaginary

Quadratic Field

4.1 Introduction

In this chapter we consider Hida families for GL2 over an imaginary quadratic field. As a

byproduct we are led to a method to exhibit torsion in the first homology groups of certain

sheaves on the 3-manifolds associated to such forms.

To explain our results let K be an imaginary quadratic field which we shall assume has

class number one (though this is almost certainly unnecessary) and let O denote its ring

of integers. We fix Kac ⊂ C and Kac ⊂ Cp. Let p be an odd rational prime which splits

in K, and π the prime of K above p corresponding to Kac ⊂ Cp. By an ordinary cuspidal

eigenform of “weight” n, levelM , and character χ : (O/MO)× → (Kac)×, we shall mean an

eigenvalue of the Hecke operators Tn acting on the corresponding space of cusp forms (see

section 4.2, but note that “weight n” is the “weight” corresponding to a sheaf of dimension

(n + 1)2, and differs by a shift of two from the normal terminology in the case of elliptic

modular forms.) with the eigenvalue of Tp prime to π. We shall write M = Npr, with N

96

prime to p and decompose χ = χac × χcy corresponding to (O/MO)× = A×B where:

A = α ∈ ((1 +NpO)/(1 +NprO)) | αα = 1 × (O/NpO)×

B = α ∈ ((1 +NpO)/(1 +NprO)) | α = α

(the anti-cyclotomic and cyclotomic parts respectively). We shall consider Λ-adic eigenforms

which interpolate ordinary cuspidal eigenforms. In the simplest case an Λ-adic eigenform of

level N (prime to p) and character φ (an anti-cyclotomic character of defined modulo Npr

for some r) is a collection am(T ) ∈ Oπ[[T ]] for each m ∈ O such that for each cyclotomic

character ψ and non-negative rational integer n:

am((1 + T )− (1 + p)2nψ(1 + p))

are the eigenvalues of the Hecke operators Tm acting on an ordinary cuspidal eigenform of

weight n, level Nps (some s) and character φψω−n where ω is the Teichmuller lifting of the

norm map (O/pO)× → (Z/pZ)× to a map (O/pO)× → Z×. In general we must allow the

ai to lie in a finite extension of Oπ[[T ]].Our main theorem (theorem 4.1) states that if we fix n then there are only finitely many

Λ-adic eigenforms of this level. If moreover we fix an anti-cyclotomic character φ then all

but finitely many ordinary cuspidal eigenforms of level Nps (any s), weight n (any n) and

character φωnψ (any cyclotomic character ψ) lift to an Λ-adic eigenform. It would be nice

to strengthen this result to say that any ordinary cuspidal eigenform lifted to a unique

Λ-adic eigenform. The problem here is torsion in the corresponding homology groups (see

the comments following theorem 4.1).

Such results have been proved by Hida (and Wiles) for modular forms over totally real

fields, however as our modular forms are not analytic we can not multiply them together

to produce new ones with good congruence properties, so we have to rely completely on

the cohomology. Our argument falls into two parts. In section 4.4 we use the inflation

restriction long exact sequence to “change level”. In section 4.5 we relate different weights.

Both are achieved by developing ideas of Hida ([Hi1]).

For example in section 4.7 we prove that if p is as above and p 6 |τ(p) where τ is Ramanu-

jam’s function ( i.e. ∆(z) =∑τ(n)e2nπiz is the cuspidal elliptic modular function of weight

97

12 for SL2(Z)), and if n1 6= n2; n1, n2 > 10; n1+n2 > 20+r; and n1 ≡ n2 ≡ 10 mod pr(p−1)then

H1(SL2(O), Sn1,n2) and H1 cusp(SL2(O), Sn1,n2)

have torsion of exponent divisible by pr. (Here Sn1,n2 is the SL2(O)-module which is the

the tensor product of the nth1 symmetric power of O2 with the natural SL2(O) action and

the nth2 symmetric power of O2 with SL2(O) action twisted by complex conjugation. Also

see section 4.2 for the meaning of “cusp” in this context.)

The rest of the paper is organised as follows. Section 4.2 contains some analytic results

we need. Section 4.3 is somewhat technical and is needed to show that our use of the

inflation restriction sequence respects the cuspidal cohomology. (Torsion prevents us using

the analytic theory of Eisenstein series which Hida used in [Hi1].) In section 4.6 we complete

the proof of our main theorem and show how to construct some examples.

We should mention that while all the theory goes through in the case of a prime inert

in K, we lack any examples in that case, so that theory may be vacuous.

Notation

Most of the notation used is either standard or explained in the text. If F is a field F ac

will denote its algebraic closure. Cp will denote the completion of Qacp and Op the ring of

elements of non-negative valuation in Cp. If π is a prime in a number field we shall use Cπ

and Oπ for the corresponding notions. If A is a ring with ideal I then we shall set:

• Γ(I) = α ∈ SL2(A) | α ≡

1 0

0 1

mod I

• Γ1(I) = α ∈ SL2(A) | α ≡

1 ∗

0 1

mod I

• Γ0(I) = α ∈ SL2(A) | α ≡

∗ ∗

0 ∗

mod I

98

We hope the context will always make it clear which ring we are talking about.

By a cofinite Zp module we shall mean the Pontriagin dual of a finite Zp module.

4.2 Review of Cohomology Groups and Automorphic Forms

Throughout let K be an imaginary quadratic field of class number one and let O denote its

ring of integers. The assumption that the class number is one is almost certainly unnecessary

but it simplifies the notation. Fix also an odd rational prime p, which is unramified in K

and a prime π of O lying above p. Note that these conditions imply that the only unit of

O congruent to 1 modulo π is 1 itself. Let¯denote complex conjugation.

For any pair of non-negative integers n1, n2 we have a free

(n1 + 1)(n2 + 1) -dimensional O module with an action of GL2(O) (or in fact of M2(O)).Itmay be explicitly described as Sn1(O2)×Sn2(O2) where Sn denotes the n-th symmetric

power (i.e. the maximal symmetric quotient of the n-th tensor power) and where γ∈GL2(O)acts on the first O2 in the natural fashion and on the second via γ. We will denote this mod-

ule Sn1,n2 . If A is an O module Sn1,n2(A) will denote Sn1,n2⊗OA. In particular Sn1,n2(C)

can be thought of as the irreducible finite dimensional representations of Lie group SL2(C).

When we need to take an O-basis we shall always take the natural basis with respect to

which γ =

a b

c d

∈M2(O) acts as:

an1M n1an1−1bM . . . bn1M

an1−1cM (an1−1d+ (n1 − 1)an1−2c)M . . . bn1−1dM

. . . .

. . . .

. . . .

cn1M n1cn1−1dM . . . dn1M

99

where M denotes the block:

an2 n2an2−1b . . . bn2

an2−1c (an2−1d+ (n2 − 1)an2−2c) . . . bn2−1d

. . . .

. . . .

. . . .

cn2 n2cn2−1d . . . dn2

We shall be interested in the cohomology of congruence subgroups

Γ < SL2(O) with coefficients in Sn1,n2(A).When A = C these groups can be studied an-

alytically. More precisely let Z denote “the quaternion upper half plane” or “hyperbolic

3-space”, that is to say:

quaternions z = x+ yk|x ∈ C and y ∈ R>0

Then SL2(C) acts on Z and in fact on Z × Sn1,n2(A) by:

γ =

a b

c d

: (z, v) 7→ ((az + b)(cz + d)−1, γv)

If Γ is torsion free, Γ\Z is a smooth manifold with a sheaf Sn1,n2(A) consisting of Γ-

invariant sections of Z × Sn1,n2(A). Then it is known that H•(Γ, Sn1,n2(A)) =

H•(Γ\Z, Sn1,n2(A)). In the case A = C this group is well studied. See for example Harder

[Ha1],[Ha2] for the following results.

There is a compact manifold with boundary Γ\Z and an embedding Γ\Z → Γ\Z which

is a homotopy equivalence (the Borel-Serre compactification). The sheaves Sn1,n2(A) extend

to Γ\Z in such a way that H•(Γ\Z, Sn1,n2(A))∼= H•(Γ\Z , Sn1,n2(A)). We have a natural

map:

H•(Γ\Z , Sn1,n2(A)) −→ H•(∂(Γ\Z), Sn1,n2(A))

We shall denote the kernel of this map by H•cusp(Γ\Z, Sn1,n2(A)) and the image by

H•Eis(Γ\Z, Sn1,n2(A)).

We have that:

100

• H icusp(Γ\Z, Sn1,n2(C)) = 0 unless n1 = n2 and either i = 1 or 2

• H1cusp(Γ\Z, Sn,n(C)) ∼= H2

cusp(Γ\Z, Sn,n(C)) ∼= Sn(Γ,C)

• H0Eis(Γ\Z, Sn1,n2(C)) = 0 unless n1 = n2 = 0 when it is equal to C

• dim H1Eis(Γ\Z, Sn1,n2(C)) =

12 dim H1(∂(Γ\Z), Sn1,n2(C))

• H2Eis(Γ\Z, Sn1,n2(C)) = H2(∂(Γ\Z), Sn1,n2(C)) unless n1 = n2 = 0 in which case it is

of codimension one

Here Sn(Γ,C) = ⊕ρU(Γ)f where U(Γ) is the closure of Γ in SL2 of the finite adeles of K

and where the sum is taken over all cuspidal automorphic representations ρ = ρf ⊗ ρ∞ of

SL2 over K with ρ∞ the principal series representation of SL2(C) corresponding to the

character

a ∗

0 a−1

7→

(a|a|

)2(n+1).

This set up may be described in terms of group cohomology as follows. Define a (Γ-)

cusp to be a Γ-conjugacy class of Borel subgroups of SL2(K). For B such a Borel we will

denote by [B] (or if necessary [B]Γ) the corresponding cusp. Set ΓB = Γ ∩B. The connected

components of the Borel-Serre compactification are in one-to-one correspondence with the

cusps. Set:

H•∂(Γ, Sn1,n2(A)) =⊕

[B]

H•(ΓB , Sn1,n2(A))

This appears to depend on the choice of Borel which represents each cusp, however if γ ∈ Γ

then ΓγBγ−1 = γΓBγ−1 and we get a canonical isomorphism

γ∗ : H•(ΓB , Sn1,n2(A))

∼−→ H•(ΓγBγ−1 , Sn1,n2(A))

Using the facts that ΓB is its own normaliser in Γ and that if M is a

G-module and g ∈ G then the map g∗ : H•(G,M)→ H•(G,M) induced by conjugation

by g on G and by translation by g on M is the identity; we see further that:

• γ∗ : H•(ΓB , Sn1,n2(A))∼−→ H•(ΓγBγ−1 , Sn1,n2(A)) is independent of the choice of γ ∈ Γ

conjugating B to γBγ−1

101

• The diagram:

H•(ΓB , Sn1,n2(A))

resրH•(Γ, Sn1,n2(A)) ↓ γ∗

resցH•(ΓγBγ−1 , Sn1,n2(A))

commutes.

We will identify the groupsH•(ΓB , Sn1,n2(A)) for B representing a cusp [B] and write simply

H•(Γ[B], Sn1,n2(A)). Restriction gives a well defined mapH•(Γ, Sn1,n2(A))→ H•∂(Γ, Sn1,n2(A))

and we have a commutative diagram:

H•(Γ, Sn1,n2(A))res→ H•∂(Γ, Sn1,n2(A))

‖≀ ‖≀H•(Γ\Z, Sn1,n2(A)) → H•(∂(Γ\Z), Sn1,n2(A))

We shall use H•cusp(Γ,M)andH•Eis(Γ,M) in the obvious way.

If we use the group cohomology the analytic description of

H•cusp(Γ, Sn1,n2(C)) given above remains true for Γ with torsion. To see this choose ∆ Γ

of finite index and without torsion. The results for Γ follow from those for ∆ because, as

we are working over a field of characteristic 0, the Serre-Hoschild spectral sequence implies

that the Γ cohomology is just the ∆\Γ invariant part of the ∆ cohomology.

We now want to describe the action of the Hecke operators on these various spaces.

For this we will work in the category of (M2(O) ∩GL2(K))-modules. It is easily seen by

abstract nonsense that for Γ a group contained in the semi-group (M2(O) ∩GL2(K)) that

the cohomology functors H•(Γ, ) defined on (M2(O) ∩GL2(K))-modules can be thought of

as the right derived functors of the fixed point functorM 7→MΓ and as such are a universal

δ-functors. This will be very helpful in checking that diagrams commutes. When we can

consider the maps as special instances of natural transformations between such universal

δ-functors, it will do to check the commutativity only in degree zero.

102

To describe the Hecke operators let g∈M2(O), det g 6= 0 and let Γ1,Γ2 be congru-

ence subgroups of SL2(O). Then we have [Γ2 : Γ2 ∩ gΓ1g−1] <∞ and so we can define a

map [Γ2gΓ1] : H•(Γ1,M)→ H•(Γ2,M) ,or more precisely a natural transformation between

δ-functors [Γ2gΓ1] : H•(Γ1, )→ H•(Γ2, ) as follows:

H•(Γ1,M)g∗−→ H•(Γ2 ∩ gΓ1g

−1,M)cor−→ H•(Γ2,M)

where the first map is induced by the compatible maps :

Mg−→ M

Γ1conjugation←− Γ2 ∩ gΓ1g

−1

and the second map is corestriction. One can check straightforwardly that [Γ2gΓ1] only

depends on the double coset Γ2gΓ1 and not on the particular choice of g. To describe

explicitly the action of [Γ2gΓ1] in degree zero and one, assume that Γ2 = ∐γi(Γ2 ∩ gΓ1g−1)

(which is easily seen to be equivalent to Γ2gΓ1 = ∐γigΓ1). Then

• [Γ2gΓ1] :MΓ1 →MΓ2 by m 7→

∑(γig)m

• [Γ2gΓ1] : H1(Γ1,M)→ H1(Γ2,M) is induced by sending a Γ1-cocycle φ to the

Γ2-cocycle δ 7→∑(γig)φ((γig)

−1δ(γjig)) where ji is the unique index such that

γ−1i δγji ∈ gΓ1g−1.

We can describe the Hecke operators on a topological level by considering the diagram:

Γ2 ∩ gΓ1g−1\Z

ւ ցΓ1\Z Γ2\Z

induced by

Zg−1 ւ ց Id

Z Z

This gives rise to:

H•(Γ2 ∩ gΓ1g−1\Z, M )

ր ց transfer

H•(Γ1\Z, M ) H•(Γ2\Z, M )

which is exactly the Hecke operator [Γ2gΓ1]. (Here M is the sheaf constructed from M

exactly as Sn1,n2 was from Sn1,n2 .)

103

We also want to define their action on the cohomology of the boundary. The topological

picture shows us how to do this. For each cusp [C] of Γ2 ∩ gΓ1g−1 we have a map:

H•(Γ1 [g−1Cg],M)g∗−→ H•((Γ2 ∩ gΓ1g

−1)[C],M)cor−→ H•(Γ2 [C],M)

(the corestriction map is defined because [Γ2 [C] : (Γ2 ∩ gΓ1g−1)[C]] is less than

[Γ2 : Γ2 ∩ gΓ1g−1]). Summing these maps over [C] we get:

[Γ2gΓ1] : H•∂(Γ1,M) −→ H•∂(Γ2,M)

A certain amount of care is needed to keep track of the identification we are making of

H•(ΓB ,M) for different Borels B representing the same cusp. One also has to check that

it is well defined up to these identifications. Moreover one can easily check that it depends

only on the double coset Γ1gΓ2 and that it is compatible with the restrictions maps from

H•(Γi,M) and the previous definition at this level. From this we see that Hecke operators

preserve the cuspidal and Eisenstein comohology. For example to check the compatibility

with our previous definition we must check that

⊕

[B] H•(Γ1 [A],M) →

⊕

[C] H•(Γ1 [g−1Cg],M)

g∗→

⊕

[C] H•((Γ2 ∩ gΓ1g

−1)[C],M)

տ ↑ ↑

H•(Γ1,M)g∗→ H•((Γ2 ∩ gΓ1g

−1),M)

and⊕

[C] H•((Γ2 ∩ gΓ1g

−1)[C],M)cor→

⊕

[C] H•(Γ2 [C],M) →

⊕

[B] H•(Γ [B],M)

↑ ր

H•((Γ2 ∩ gΓ1g−1),M)

cor→ H•(Γ2,M)

commute. The first diagram is easy. That the pentagon commutes follows from the

following fact,which it suffices to check in degree zero:

Assume Γ ⊃ ∆ with finite index and Γ ⊃ C and M is a Γ-module. Let C1, ..., Cs

be representatives of the ∆-conjugacy classes of Γ-conjugates of C. Then

⊕H•(∆ ∩ Ci,M)

cor→⊕H•(Ci,M)

conjugation→ H•(C,M)

↑ res ↑ resH•(∆,M)

cor−→ H•(Γ,M)

commutes.

104

To consider the action of the Hecke operators analytically we must modify our analytic

description slightly. Let U be an open compact subgroup of GL2(Af ) (here Af denotes the

finite adeles of K) such that Γ = U ∩GL2(K) ⊂ SL2(O). Then we have:

• Sn(Γ,C) = ⊕ρUf where the sum is taken over all cuspidal automorphic representa-

tions ρ = ρf ⊗ ρ∞ of GL2(K) with ρ∞ the principal series representation of GL2(C)

corresponding to the character

a ∗

0 b

7→

(a|a|

)n+1 ( |b|b

)n+1|ab|−n.

• H1Eis(Γ\Z, Sn1,n2(C)) = Gn1,n2(Γ,C) =

⊕(Ind

GL2(Af )

B0(Af )ψf

)Uwhere the sum is taken

over all Hecke characters:

ψ = ψfψ∞ :

A× ∗

0 A×

→ C×

for which ψ∞ :

a ∗

0 b

7→ aa−n2b−(n1+1). (The induction here is not the usual

unitary induction but that in Harder [Ha2], which explains the slight discrepancy

from the cuspidal case above.)

Now assume U1, U2 are as above and that g ∈ GL2(K), then we get a map:

[U1g−1U2] : Sn(Γ1,C) = ⊕ρU1

f −→ ⊕ρU2f = Sn(Γ2,C)

by v 7→∫(U1g−1U2)

v ρf (x) dx where the Haar measure dx is normalised so that∫U1dx = 1. If

Γ2gΓ1 = ∐γugΓ1 then U1g−1U2 ⊃ ∐U1g

−1γ−1u . If this is in fact an equality then [U1g−1U2]

and [Γ2gΓ1] : Sn(Γ1,C) → Sn(Γ2,C) coincide. This follows for example from results in

Harder [Ha2]. Similar results hold for the Eisenstein cohomology.

Finally in this section we consider some special Hecke operators. Let T be the abstract

commutative ring over Z generated by the symbols Tn for n ∈ O \ 0. For the rest of this

section we consider only congruent subgroups of the form:

a b

c d

∈ SL2(O) | c ≡ 0 modN , a ≡ d ≡ 1 modM

105

whereM | N . For such a group Γ, Tn 7→ [Γ

n 0

0 1

Γ] gives an action of T on H•(Γ,M) .

(To see this is a good definition one must check that these operators commute. By abstract

nonsense we can check this in degree zero where the problem reduces to showing that if

Γ

n 0

0 1

Γ = ∐αΓ and Γ

m 0

0 1

Γ = ∐βΓ then the cosets αβΓ and βαΓ coincide.

For this we refer the reader to Shimura [Sh2].) Note that we can apply the remarks of the

last paragraph to describe these Hecke operators analytically, taking:

U =

a b

c d

∈

∏

v

GL2(Ov) | c ≡ 0 mod N , d ≡ 1 modM

(Atleast if 1 is the only unit of O congruent to 1 mod M .) Moreover we can factorise

[U1

n−1 0

0 1

U2] into a product of local operators, which are the identity for primes not

dividing n.

If Γ1 > Γ2 are both of the above form defined by N1,M1;N2,M2 and if n is not divisible

by primes dividing N1 but not N2, then we have a commutative diagram:

H•(Γ1,M)Tn−→ H•(Γ1,M)

res ↓ ↓ resH•(Γ2,M)

Tn−→ H•(Γ2,M)

and so if N1 and N2 have the same prime factors we see that the restriction map is T-

equivariant (or as we shall write “Hecke equivariant”). (This need only be checked in

degree zero where it follows because we can find γi such that Γ1 = ∐γi(Γ1 ∩ gΓ1g−1) and

Γ2 = ∐γi(Γ2 ∩ gΓ2g−1) - see Shimura [Sh2].)

We have an action of (O/NO)× ∼= Γ0(N)/Γ1(N) on H•(Γ1(N),M) by conjugation. We

shall fix Γ0(N)/Γ1(N)∼→ (O/NO)× by

a b

c d

7→ d.Then it is again easily checked that

this action commutes with that of T, and that if N | N ′ then (O/NO)× →→ (O/N ′O)× is

compatible with res : H•(Γ1(N),M)→ H•(Γ1(N′),M).

We shall be particularly interested in the Hecke operators Tµ for µ dividing some power of

p. Thus we recall the decomposition for Γ now of the form

106

a b

c d

∈ SL2(O) | c ≡ 0 modNpr , a ≡ d ≡ 1 modNps with r ≥ 1 and r ≥ s and for

µ as above:

Γ

µ 0

0 1

Γ = ∐

µ u

0 1

Γ

as u runs over any set of representatives for congruent classes of O mod µ. If ν also divides

some power of p we have also:

[Γ

µ 0

0 1

Γ][Γ

ν 0

0 1

Γ] = [Γ

µν 0

0 1

Γ]

Most essentially we have Hida’s idempotent. In general to an operator T on a finite (or

cofinite) Zp-module H we can define an idempotent eT in EndZp(H) commutes with T such

that T is an automorphism of eTH and is topologically nilpotent on (1− eT )H (see Mazur

and Wiles [MW2]). In fact eT = limr→∞Tr!. If S : H → H ′ is a morphism between finite

Zp-modules and T, T ′ are operators on H and H ′ respectively such that ST = T ′S then S

restricts to a map S : eTH → eT ′H ′. In particular for the Hecke operator Tp acting on a

finite Zp-module H we will denote the corresponding idempotent simply e, call it Hida’s

idempotent and write H for eH, which we will call the ordinary part of H. If p = ππ in Othen we can define eπ and eπ similarly corresponding to Tπ and Tπ and we have eπeπ = e.

In most of the following we will restrict to the ordinary parts of modules, and this will be

essential for our arguments. Any v ∈ H which is an eigenvector of Tp with eigenvalue a

p-adic unit will be preserved by e. We see that almost all of the discussion of this section

goes over to ordinary parts in the obvious fashion, but one should be aware that restriction

does not usually map (for example) H•(Γ1(N),M) to H•(Γ1(Np),M) when p 6 |N . (This

is “because we are going from no p in the level to p in the level”.)

Although Hida’s idempotent is initially defined only in the p-adic setting we can some-

times consider it in more general situations. Fix embeddings Kac ⊂ Cπ and Kac ⊂ C. In

particular we have fixed an extension of the π-adic valuation to Kac. Now if H is a finite

107

torsion free O-module with an action of Tp then we have:

e ∈ Oπ[Tp] ⊂ EndOπ(Hπ)

∩ ∩Kπ[Tp] ⊂ EndKπ(H ⊗Kπ)

∪ ∪K[Tp] ⊂ EndK(H ⊗K)

and there is a finite extension L/K contained in Kπ such that e ∈ L[Tp] and hence ∈EndC(H ⊗ C).

Then we have that:

• eSn(Γ1(Npr),C) =

⊕(eρ

Upp )⊗ (ρpf )

Up

• eGn1,n2(Γ1(Npr),C) =

⊕(e Ind

GL2(Kp)B0(Kp)

ψp

)Up⊗ Vψp

f

where:

Up =

a b

c d

∈ GL2(Op) | c ≡ 0 mod Npr, d ≡ 1 mod Npr

Up =

a b

c d

∈

∏

v 6 |pGL2(Ov) | c ≡ 0 mod N, d ≡ 1 mod N

here ρ = ρp ⊗ ρpf ⊗ ρ∞ runs over cuspidal automorphic representations as before, where

ψ = ψp×ψpf ×ψ∞ is also as described above, and where Vψpfdoes not matter very much. If

p = ππ in O then we have moreover, eρUpp = eπρ

Uππ ⊗ eπρUππ . It is known that for v a prime

above p with ρUvv 6= 0 and for r > 0:

• ρv supercuspidal implies that evρUvv = 0

• ρv = ρ(ψ,ψ| . |) implies that ev is undefined if ψi is unramified and (Nv)12ψ(v) has

positive π-adic valuation. Otherwise the dimension of evρUvv is 1 or 0 according to

whether ψ is unramified and (Nv)12ψ(v) is a π-adic unit, or not.

• ρv = ρ(ψ1, ψ2) principal series implies that ev is undefined if ψi is unramified and

(Nv)12ψi(v) has positive π-adic valuation for some i. Otherwise evρ

Uvv has dimension

108

0, 1 or 2 corresponding to the number of ψi (i = 1, 2) which are unramified and have

(Nv)12ψi(v) a π-adic unit.

From this one can show that if p is inert in K then eGn1,n2(Γ1(Npr),C) = 0 and that

if p = ππ splits in K then:

eGn1,n2(Γ1(Npr),C) =

⊕Vψp

f

where the sum is taken over characters ψ = ψp×ψpf×ψ∞ as described above with the added

condition that if ψp :

a ∗

0 b

7→ ψ1(a)ψ2(b) then ψ1 is unramified at π and ψ2 at π. In

particular for r ≥ 1, dim eGn1,n2(Γ1(Npr),C) is independent of n1 and n2 (as 1 is the only

unit ofO congruent to 1 mod (Npr)), and dim eGn1,n2(Γ1(Npr),C) ≥ dimGn1,n2(Γ1(N),C).

This is because the p part of the representation corresponding to ψ is the principal

series representation coming from χ1 and χ2 where in the inert case, up to roots of unity,

pχ1(p) = pn2+1 and pχ2(p) = pn1+1. While in the split case:

• p 12χ1(π) = πn2+1

• p 12χ2(π) = πn1+1

• p 12χ1(π) = πn2+1

• p 12χ2(π) = πn1+1

4.3 Cohomology of the Boundary

In this section we want to describe the ordinary part of the cohomology of the boundary.

We shall let M denote a finite or cofinite Zp-module with a continuous action of the mul-

tiplicative semi-group M2(O)∩GL2(K) in the p-adic topology. Γ will denote a congruence

subgroup of SL2(O) of the form:

a b

c d

∈ SL2(O) | c ≡ 0 mod Npr, d ≡ 0 mod Np?

109

For B a Borel we shall let ΓB denote the unipotent radical of ΓB . Then ΓB ΓB and

WB = ΓB/ΓB embeds into the roots of unity in K and so [ΓB : ΓB] is prime to p. Thus we

have:

res : H•(ΓB ,M)∼−→ H•(ΓB ,M)WB

(from the Hoschild-Serre spectral sequence).

It is known that there is a bijection:

Γ\SL2(O)/B0 ←→ Γ− cuspsγ 7−→ γB0γ

−1

where B0 =

∗ ∗

0 ∗

. Then γ−1ΓγB0γ−1γ = γ−1Γγ ∩

1 O

0 1

and we can think of

ΓB as an ideal in O. As B is its own normaliser we can easily deduce that this ideal depends

only on the cusp [B] and not on its representation γB0γ−1.We shall write Γ[B] for this ideal.

Similarly we can legitimately write W[B] and H•(Γ[B],M). Then we have canonically:

H•∂(Γ,M) ∼=⊕

[B]

H•(Γ[B],M)W[B] ∼=⊕

[B]

H•ct(Γ[B], p,M)W[B]

Here the “ ct ” indicates that we are using continuous cohomology, Γ[B], p denotes the closure

of Γ[B] in Op and the latter isomorphism follows because there is a bijection between Γ[B]

cocycles (or coboundaries) and continuous Γ[B], p cocycles (coboundaries). We shall drop

the “ ct ” from our notation in future. In the case p = ππ is split in O we see that:

H•(Γ[B],M) = H•(Γ[B], π,M)⊕H•(Γ[B], π,M)

and that the W[B] action preserves this decomposition. Thus we can write:

H•∂(Γ,M) = H•∂(Γ,M)π ⊕H•∂(Γ,M)π

We shall introduce the following definitions for a cusp [B], and forv a prime above p:

• [B] is v-unramified if Γ[B], v = Ov.

110

• [B] is v-first class if [B] has a representative γB0γ−1 with γ ∈ SL2(O) and γ ≡

∗ ∗0 ∗

mod vs for all s ( or equivalently for s = r)(1).

• [B] is v- third class if [B] has a representativeγB0γ−1 with γ =

a ∗

c ∗

where

(c, v) = 1. In this case any representative has this form(2) and we can chose a repre-

sentative with a ≡ 0 mod vs for any s(1).

• [B] is v-second class if it is neither v-first nor v-second class.

Here the results marked (1) follow as Γ ⊃ Γ1(Npr) and those marked (2) as Γ ⊂ Γ0(p). Any

first class cusp is unramified, as follows easily from the formula:

a b

c d

1 α

0 1

a b

c d

−1

=

1− acα a2α

−c2α 1 + acα

and the fact (1).

We shall prove:

Proposition 4.1 If v is any prime above p and Γ and M are as above then we have an

injection:

evH•∂(Γ,M)v →

⊕

[B]

H•(Γ[B], v,M)

where the sum is over v-first class cusps.

Before proving this we shall draw the corollary that will be of use to us later:

Corollary 4.1 If Γ1 ⊃ Γ2 are as Γ above and M is as above then the restriction map:

eH•∂(Γ1,M)res→ H•∂(Γ2,M)

is injective (on the ordinary part).

Proof: It will do to show that for each prime v above p

evH•∂(Γ1,M)v

res→ H•∂(Γ2,M)v

111

Then considering the commutative diagram:

evH•∂(Γ1,M)v

res−→ H•∂(Γ2,M)

res ↓ ↓ res⊕

[B] v−class 1H•(Γ1[B], v,M)

res→⊕

[C]H•(Γ2[C],M)

the proposition tells us that the left hand vertical arrow is injective and so we need only

show that the lower horizontal arrow is injective. But above each v-first class cusp [B] of

Γ1 there lies a v-first class cusp [C] of Γ2, as we see at once from the definition, and

H•(Γ1[B], v,M)∼−→ H•(Γ2[C], v,M)

as Γ1[B], v = Ov = Γ2[B], v.

We now turn to the proof of the proposition. We shall assume p = ππ (which is the case

of real interest and slightly harder than the inert case), and that v = π (v = π is exactly

the same).

Fix n such that n > r and πn ≡ 1 mod Nπr and set g =

πn 0

0 1

. We shall show

that:

• If x ∈ ker(H•∂(Γ,M)π →⊕

[B] class 1H•(Γ[B], π,M)) then

[ΓgΓ]x ∈⊕[B] class 3H

•(Γ[B], π,M)W[B] and this latter space is preserved by [ΓgΓ].

• [ΓgΓ] :⊕

[B] class 3H•(Γ[B], π,M)W[B] →⊕

[B] class 3H•(Γ[B], π,M)W[B] is topologically

nilpotent.

from which the proposition follows easily. To prove the second assertion it will do to take

M of finite cardinality. Pictorially this all amounts to [ΓgΓ] acting as follows:

112

cusps

third class

cusps

second class

cusps

first class

topologically

nilpotent

Let [B] be a Γ-cusp with B = δB0δ−1, δ =

a ∗

c ∗

∈ SL2(O), we want to examine the

[B]-component of [ΓgΓ]. Firstly the Γ ∩ gΓg−1 cusps above [B] are exactly [γ−1i Bγi] where

Γ =∐

ΓBγi(Γ∩ gΓg−1) or (as we see after a small calculation) ΓgΓ =∐

ΓBγigΓ. Thus the

cusps [C] such that [ΓgΓ] gives a non-zero map from H•(Γ[C],M) to H•(Γ[B],M) are repre-

sented by [g−1γ−1i Bγig], or equivalently by [g−1u Bgu] where gu =

πn u

0 1

as u varies over

congruence classes mod πn and without loss of generality u ≡ 0 mod Nπr. Explicitly [C]

is represented by εB0ε−1 with ε =

a−ucπm ∗

πn−mc ∗

∈ SL2(O) for some m. Thus if [B] is not

third class

( i.e. if π|c) then we see that π 6 |(a − uc) so that m = 0 and [C] must be class one,

which is our first claim.

For the second assertion consider [B] a third class cusp, which we can write [δB0δ−1]

with δ =

π2na b

c d

where π 6 |d. Then if [C] is a third class cusp giving rise to [B] we

must have u ≡ 0 mod πn ( u as above) and so we may take u = 0 and C = εB0ε−1 with

113

ε =

πna b

c πnd

. Then λ ≡ 0 mod Nπr and cλ ≡ d mod πr implies that:

ε

1 λ

0 1

∈ Γ1(Np

r)

Thus [B] = [C] and the map:

[ΓgΓ] :⊕

[B] class 3

H•(Γ[B], π,M) −→⊕

[B] class 3

H•(Γ[B], π,M)

splits up as a direct sum of maps:

H•(∆1π,M)g∗−→ H•(∆3π,M)

cor−→ H•(∆2π,M)

where:

• ∆1 = ΓεB0ε−1 =

1− πnacα π2na2α

−c2α 1 + πnacα

∈ Γ | α ∈ O

• ∆2 = ΓδB0δ−1 =

1− π2nacα π4na2α

−c2α 1 + π2nacα

∈ Γ | α ∈ O

• ∆3 = ∆2 as ∆2 ⊂ gΓg−1

The map g∗ is induced by the compatible maps M →M , m 7→ gm and ∆3 → ∆1 by:

1− π2nacα π4na2α

−c2α 1 + π2nacα

7−→

1− πnac(πnα) π2na2(πnα)

−c2(πnα) 1 + πnac(πnα)

and cor reduces to the identity.

Thus we may describe [ΓgΓ] : H•(Γ[B], π,M) → H•(Γ[B], π,M) as the map induced by

compatible maps M →M by m 7→ m and Γ[B], π → Γ[B], π by α 7→ πnα. Then if M is finite

(as a set), for some a, res : H•(Γ[B], π,M)→ H•(πanΓ[B], π,M) is zero, and so by our above

description T aπn is also zero, which is what we wanted to show.

114

4.4 Change of Level

The central result of this section is:

Proposition 4.2 LetN ∈ O be prime to p, let r ≥ s ≥ 1 and let M be a Zp-module. Set

Gr,s = (1 +NpsZ)/(1 +NprZ). Then

i) res : eH•(Γ1(Nps),M)

∼−→ eH•(Γ1(Nps) ∩ Γ0(Np

r),M)

If further eMΓ1(Npr) = 0 then

ii) res : eH1(Γ1(Nps) ∩ Γ0(Np

r),M)∼−→ eH1(Γ1(Np

r),M)Gr,s

and hence

iii) eH1(Γ1(Nps),M)

∼−→ eH1(Γ1(Npr),M)Gr,s

The modules M = Sn1,n2(A) satisfy eMΓ1(Npr) = 0 for A any Oπ-module.

Proof: Let us establish the notation:

Γs = Γ1(Nps)

⋃

Φ = Γ1(Nps) ∩ Γ0(Np

r)

Γr = Γ1(Np

r)

Then Φ/Γr ∼= Gr,s.

i) I claim the following diagram commutes:

H•(Γs,M)res−→ H•(Φ,M)

T r−sp ↓ ւ ↓ T r−sp

H•(Γs,M)res−→ H•(Φ,M)

where the diagonal arrow is given by the Hecke operator [Γs

pr−s 0

0 1

Φ]. From this the

result would follow at once as Tp is invertible on the ordinary part of any module. Further

it suffices to check the commutativity in degree 0 by abstract nonsense, and here it follows

from the existence of γu such that:

115

• Γs

pr−s 0

0 1

Γs = ∐γuΓs

• Γs

pr−s 0

0 1

Φ = ∐γuΦ

• Φ

pr−s 0

0 1

Φ = ∐γuΦ

(see Shimura [Sh2]).

ii) We look at the inflation-restriction sequence:

0→ H1(Gr,s,MΓr)

inf→ H1(Φ,M)res→ H1(Γr,M)Gr,s

t→ H2(Gr,s,MΓr)

Let Tp and hence e act on these groups by giving them their normal action on the middle

terms and letting them act on the outer terms through their action on MΓr = H0(Γr,M).

Assuming for a minute that these actions are compatible we see that there is an exact

sequence:

0 −→ eH1(Φ,M)res−→ eH1(Γr,M)Gr,s −→ 0

as desired.

To check the compatibility assertion let¯denote the map:

Φ/Γr∼−→ Gr,s

a b

c d

7−→ d

Let also gu be such that:

Γr

p 0

0 1

Γr = ∐guΓr and Φ

p 0

0 1

Φ = ∐guΦ

For γ, δ ∈ Φ let v = v(u) and w = w(u) be the unique indices such that g−1u γgv and

g−1v δgw ∈ Φ. It is easily checked that g−1u γgv = γ (and that g−1v δgw = δ). Then:

a) Let φ ∈ Z1(Gr,s,MΓr ). Then for γ ∈ Γr:

116

• inf(Tpφ)(γ) = (∑guφ) γ

• (Tp inf φ)(γ) =∑guφ(g

−1u γgv)

and so inf(Tpφ) = Tp(inf φ).

b)Let x ∈ H1(Γr,M)Gr,s . Then there exists φ ∈ C1(Φ,M) such that:

• φ |Γr∈ Z1(Γr,M) and represents x

• dφ ∈ Z2(Gr,s,MΓr) ⊂ C2(Φ,M) and represents t(x)

(see Hoschild-Serre [HS]).

Tpt(x) is represented by (γ, δ) 7→ ∑gu(φ(γδ) − γφ(δ) − φ(γ)). Moreover consider ψ ∈

C1(Φ,M) defined by ψ(γ) =∑guφ(g

−1u γgv). Then ψ |Γr∈ Z1(Γr,M) and represents Tpx.

Moreover

(dψ)(γ, δ) =∑

u guφ(g−1u γδgw)− γ

∑u gvφ(g

−1v δgw)−

∑u guφ(g

−1u γgv)

=∑

u gu(φ(g−1u γgvg

−1v δgw)− (g−1u γgv)φ(g

−1v δgw)− φ(g−1u γgv)

)

=∑

u gu(dφ)(g−1u γgv , g

−1v δgw)

=∑

u gu(dφ)(γ, δ) as g−1u γgv = γ and g−1v δgw = δ

=∑

u gu(φ(γδ) − γφ(δ) − φ(γ))

Thus (dψ) ∈ Z2(Gr,s,MΓr) and represents Tpt(x), i.e. t(Tpx) = Tpt(x).

For the final assertion that eSn1,n2(A)Γr = 0 we shall make an arbitrary extension of Tp

to all of Sn1,n2(A) and show that:

TpSn1,n2(A) ⊂ pSn1,n2(A)

from which it follows that eSn1,n2(A) and hence eSn1,n2(A)Γr vanish. Choose a set of

representatives u for congruence classes of O mod p, and set:

Tpm =∑

u

p u

0 1

.m

117

Then:

Tpm ≡ ∑u

0 . . . . . . . 0 un1 un2

0 . . . . . . . 0 un1 un2−1

. . .

. . .

0 . . . . . . . 0 un1

0 . . . . . . . 0 un1−1un2

. . .

. . .

0 . . . . . . . 0 un1−1

. . .

. . .

. . .

0 . . . . . . . 0 1

m mod π

≡ 0m mod π

as for any character χ : (O/πO)× → (O/πO)× one has:

∑

u∈(O/πO)×

χ(u) = 0

. Q.E.D.

Now define Hn1,n2= lim→ r

eH1(Γ1(Npr), Sn1,n2(Kπ/Oπ)). Then Hn1,n2

is an Oπ[[G]]-module where G = lim

←Gr with Gr = (O/NprO)×. Let Hr = ker(G →→ Gr) then what we

have just shown amounts to:

(Hn1,n2)Hr = eH1(Γ1(Np

r), Sn1,n2(Kπ/Oπ)) (r ≥ 1)

Ifˇdenotes the Pontriagin dual this is the same as:

(Hn1,n2)Hr = eH1(Γ1(Np

r), Sn1,n2(Oπ))

We have the following decompositions for G:

G = Gtor ×H1 = (O/NO)× ×H0 = (O/NO)× × (O/pO)× ×H1

Moreover complex conjugation acts on H1 and we have H1 = H+1 ×H−1 where ± refer to the

corresponding eigenspaces for complex conjugation. We can choose isomorphisms Zp → H±1

118

by λ 7→ uλ±. Then

Oπ[[G]] ∼= Oπ[[H1]][Gtor ] ∼= Oπ[[T+, T−]][Gtor ]

the latter map being given by u± ↔ (T±+1). Oπ[[H1]] is a complete local noetherian ring,

its maximal ideal corresponds to (π, T+, T−)Oπ[[T+, T−]].

Corollary 4.2 Hn1,n2is a finitely generated Oπ[[H1]]-module.

Proof: By Nakayama’s lemma it will do to show that Hn1,n2is compact and that (Hn1,n2

)H1 =

(Hn1,n2/(T+, T−)Hn1,n2

) is finitely generated over Oπ. The proposition and the fact that

Hn1,n2= lim

←eH1(Γ1(Np

r), Sn1,n2(Oπ)) reduces this to the well known fact that

H1(Γ1(Npr), Sn1,n2(Oπ)) is a finitely generated Oπ-module.

From the corollary of section 4.3 we have for r ≥ s ≥ 1 a commutative diagram with

exact rows:

0 → eH1cusp(Γ1(Nps),M) → eH1(Γ1(Nps),M) → eH1

Eis(Γ1(Nps),M) → 0

↓ ↓≀ ↓

0 → eH1cusp(Γ1(Npr),M)Gr,s → eH1(Γ1(Npr),M)Gr,s → eH1

Eis(Γ1(Npr),M)Gr,s

where the right hand vertical arrow is an injection and where M = Sn1,n2(Kπ/Oπ). Thus

the left hand vertical arrow is an isomorphism, and if we set Hn1,n2 cusp to be

lim→

eH1cusp(Γ1(Np

r), Sn1,n2(Kπ/Oπ)) then we see that:

• Hn1,n2 cusp → Hn1,n2

• Hn1,n2 cusp is a finitely generated Oπ[[H1]]-module.

• (Hn1,n2 cusp)Hr = eH1

cusp(Γ1(Npr), Sn1,n2(Kπ/Oπ)) for r ≥ 1.

4.5 Change of Weight

For n1, n2 integers set:

µn1,n2 : G = (O/NO)× ×H0 → H0 = O×p → O×p → O×πα 7→ αn1αn2

119

This extends to a homomorphism µn1,n2 : Oπ[[G]] → Oπ and to a homomorphism µn1,n2 :

Oπ[[G]]→ Oπ[[G]] defined by sending h ∈ G to µn1,n2(h)h. Then

Oπ[[G]]µn1,n2−→ Oπ[[G]]

µn1,n2 ց ւ µ0,0

Oπ

commutes. ForM an Oπ[[G]]-module we define a twistM(µn1,n2) to be the same underlying

topological abelian group but with a new Oπ[[H0]] action defined by:

h.m = µn1,n2(h)m

Our aim is to prove:

Proposition 4.3 There is a canonical Hecke equivariant isomorphism:

Hn1,n2

∼−→ H0,0(µn1,n2)

It also restricts to an isomorphism on the cuspidal parts.

Before proving this we note a couple of corollaries. From now on we will use simply H

to denote H0,0.

Corollary 4.3 Let I be the closed ideal of Oπ[[H1]] generated by

h−µn1,n2(h) | h ∈ Hr then H/IH ∼= eH1(Γ1(npr), Sn1,n2(Oπ)), and a similar statement

holds for the cuspidal part.

This is clear.

Corollary 4.4 Hcusp is a torsion Oπ[[H1]]-module.

Proof: Let Ir denote the closed ideal generated by h− µ0,1(h) | h ∈ Hr and let R denote

Oπ[[H1]] and M denote H. Then we have that M/IrM is p-torsion and that R/Ir is free

of p-torsion. If M were not a torsion R-module we would have an injection R φ→M . Let N

denote the submodule n ∈M | ∃r ∈ Rwith rn ∈ Imφ. Then we can define θ : N → FR

120

such that θ φ = IdR and there is then a non-zero constant λ ∈ R such that λIm θ ⊂ R.Then we have for each r:

M/IrM ← N/IrN →→ λ Imθ/Irλ Imθ →→ (λ Imθ + Ir)/Ir ← ((λ) + Ir)/Ir → R/Ir

Now M/IrM is p-torsion and thus ((λ) + Ir)/Ir is p-torsion and so in fact zero. Thus

λ ∈ Ir ∀r which implies λ = 0 a contradiction.

We now turn to the proof of the proposition. We have that:

Hn1,n2= lim→eH1(Γ1(Np

r), Sn1,n2(O/πrO))

where the ith term on the right hand side is an O/πrO[(O/Npr)×]-module. Call this ring

Λr, then Oπ[[G]] = lim←

Λr and this is compatible with the above direct limit. µn1,n2 reduces

to a map Λr → O/πrO.Let

j : Sn1,n2(O/πrO) −→ O/πrO

x0

.

.

xn1n2

7−→ xn1n2

where we choose a basis of Sn1,n2(O/πrO) as described at the start of section 4.2. It is a

map of modules over the semi-group:

∆ =

a b

c d

∈M2(O) | ad− bc 6= 0, c ≡ 0 mod pr, d ≡ 1 mod pr

It thus induces a Hecke equivariant map:

j∗ : H•(Γ1(Np

r), Sn1,n2(O/πrO)) −→ H•(Γ1(Npr),O/πrO)

and for r ≥ s ≥ 1:

H•(Γ1(Nps), Sn1,n2(O/πrO))

j∗−→ H•(Γ1(Nps),O/πrO)

res ↓ ↓ res

H•(Γ1(Npr), Sn1,n2(O/πrO))

j∗−→ H•(Γ1(Npr),O/πrO)

121

commutes. It is easily checked that j∗ gives a map of Λr-modules:

H•(Γ1(Npr), Sn1,n2(O/πrO)) −→ H•(Γ1(Np

r),O/πrO)(µn1,n2)

(In fact it suffices to check that for σ ∈ (O/prO)× we have

j∗ σ = σn1 σn2(σ j∗) and such an equality need only be checked in degree zero where it

really is easy.)

Finally we know from lemma 1.1 that j∗ is an isomorphism on ordinary parts as j :

gSn1,n2(O/πrO)∼→ g(O/πrO) for g =

pr 0

0 1

.

4.6 Cyclotomic Hida Families

First let us introduce a definition of an eigenform ( of the Hecke operators having weight

n, level Npr, and character χ : (O/NprO)× → Kac×) suited to our purposes. First note

that it is easy to see that there are natural bijections between the homomorphisms of each

of the following forms:

• λ : T(H1(Γ1(Npr), Sn,n(Kπ/Oπ))χ)→ Oπ

• λ : T(H1(Γ1(Npr), Sn,n(Oπ))χ)→ Oπ

• λ : T(H1(Γ1(Npr), Sn,n(Kπ))χ)→ Cπ

• λ : T(H1(Γ1(Npr), Sn,n(Kπ))

χ)→ Cπ

• λ : T(H1(Γ1(Npr), Sn,n(Oπ))χ)→ Oπ

• λ : T(H1(Γ1(Npr), Sn,n(Kπ/Oπ))χ)→ Oπ

• λ : T(H1(Γ1(Npr), Sn,n(C))χ)→ C ( using our embedding of Kac into Cπ and C and

the fact that the eigenvalues of the Hecke operators are algebraic)

We will call such a homomorphism an eigenform of weight n, level Npr , and character

χ. We will call it ordinary (respectively cuspidal) if it factors through the Hecke algebra

122

acting on the ordinary (respectively cuspidal) part of the cohomology. Thus the cuspidal

eigenforms correspond to homomorphisms:

• λ : T(Sn(Γ1(Npr),C)χ)→ C

Let G,H1, etc. be as in the last section. Let Λ = Oπ[[H+1 ]] and let R denote the integers

in the algebraic closure of FΛ. If χ : Gtor ×H− → O×π is a finite character we shall mean

by an (ordinary) Λ-adic eigenform of level N and character χ a homomorphism:

λ : T(Hχ) −→ R

or equivalently:

λ : T(Hχ) −→ R

We shall call it cuspidal if it factors through Hχcusp.For ψ : H1 → O×π a finite character and n a positive integer set:

ψn : H1 −→ O×π

h 7−→ h2nψ(h)

and denote also its extension to a map Λ→ Oπ by ψn. Let ℘ψ,n be the kernel of ψn and Qa prime of R above ℘ψ,n. Also let ω : (O/pO)× → µ(Oπ) denote the unique lifting of the

norm map (O/pO)× → (Z/pZ)× ( i.e. the Teichmuller character). Then we have:

T(Hχ) −→ R↓ ↓

T(eH1(Γ1(Npr), Sn,n(Oπ))χψω−n) = T(Hχ/℘ψ,nHχ) − → R/Q ⊂ Oπ

where the vertical arrows are surjective and the map across the bottom making the diagram

commute exists and is unique. (It exists because if I is the kernel of the vertical map on

the left then I ⊂ √℘ψ,n (as Hχ is finitely generated) and√℘ψ,n ⊂ ker(T(Hχ) → R/Q)

because ℘ψ,n is contained in this kernel.) This just says that for each ψ, n,Q as above λ

reduces modulo Q to a unique ordinary eigenform of weight n, level Npr for sufficiently

large r, and character χψω−n. A similar statement is true for cuspidal Λ-adic eigenforms.

We will prove a partial converse to this.

123

Theorem 4.1 Fix N ∈ O. Then :

1. There are only finitely many cuspidal Λ-adic eigenforms of level N .

2. If we also fix a finite character χ : Gtor×H− → O×π then for almost all ordinary eigen-

forms λ of level Npr, weight n and character ψχ for some r, n and ψ a cyclotomic

character

( i.e. ψ : (1 + pO/1 + prO) → O×π and ψ(α) = ψ(α)) there is a Λ-adic eigenform λ

of level N , a character χωn and a prime Q of R above ℘ψ,n such that λ reduces to λ

modulo Q.

3. For all but finitely many finite characters χ : Gtor ×H− → O×π there are only finitely

many cuspidal eigenforms of weight n, level Nps, and character χψ as s, n and ψ

vary, with ψ cyclotomic.

Remark It would be very nice to be able to strengthen part 2 to assert that we could

lift any ordinary cuspidal eigenform to a unique Λ-adic one. This is false if one does

not restrict to the cuspidal part as the ordinary Eisenstein series come in two variable p-

adic families. However the examples of cuspidal forms coming either by base change from

GL2/Q or by theta series from grossencharacters on a quadratic extension of K fit nicely

into cyclotomic families. One could prove this strengthening if one knew that Hcusp was

a free Λ-module. This latter statement is equivalent to eH1 cusp(Γ1(Np), Sn,n(Oπ)) being

torsion free for infinitely many n. This is a consequence of the following result, which in

turn follows easily from Nakayama’s lemma and unique factorisation:

If πi ∈ Λ are infinitely many distinct primes (all different from p) and M is a

compact Λ-module then the following are equivalent:

• M is a free Λ-module

• for all i M/πiM is p-torsion free.

Proof: Note that part 3 follows from 1 and 2.

1) First note that Hcuspχ is a finite torsion Λ-module for all but finitely many χ and that

124

there are no Λ-adic eigenforms for such χ. (Infact Oπ[[G]] has only finitely many height

one primes P such that HcuspP 6= 0 (such a prime is a minimal element of the support of

Hcusp) and so only finitely many such that Oπ[[G]]/P → Hcusp/PHcusp.) For the finite set

of χ for which Hcuspχ is not a torsion Λ-module it is finite and hence T(Hcuspχ) is a finite

Λ-algebra so there are only finitely many Λ-adic eigenforms of character χ.

2) Let T = T(Hχωm) for some m = 0, . . . p − 2. Then for n ≡ m mod (p − 1)

T →→ T(H1(Γ1(Npr), Sn,n(Oπ))χψ), and an ordinary eigenform λ of weight n character

χψ can be thought of as a map T λ→ Oπ. If ker λ is not a minimal prime ideal then there

is a prime P such that ker λ strictly contains P and such that T /P → R and so λ can be

lifted. But T has only finitely many minimal prime ideals (as it is finite over an integral

domain) and so there are only finitely many maps T λ→ Oπ whose kernel is a minimal prime

of depth 2. This implies that there are only finitely many eigenforms λ as in the theorem

of weight n ≡ m mod (p − 1) which can not be lifted to a Λ-adic eigenform, which at once

implies the result.

Finally in this section we would like to give some examples to show that our theory is

not vacuous. Although one could describe these examples very precisely we shall content

ourselves with an existence theorem.

Proposition 4.4 Assume that p = ππ is split in K then:

1. Let N ∈ Z be prime to p and such that there is an ordinary cuspidal eigenform of

weight k ≥ 2 and level Npr for some r for GL2/Q. Then there is a cuspidal Λ-adic

eigenform over K of level N .

2. If χ : (O/prO)× → C×π is an anticyclotomic character

(i.e. χ(α) = χ(α)−1 for α ∈ (1 + pO) ) then there is an integer N prime to p

and a character ′χ : (O/NO)× → C×π and a Λ-adic eigenform over K of level N and

character χ′χ.

Proof: 1) This comes from base change.

More precisely if the assumptions of the proposition hold then we know from the work

of Hida (see for example [Hi1]) that as r varies there are infinitely many ordinary cuspidal

125

eigenforms of level Npr and weight k for GL2/Q. Hence there are infinitely many cuspidal

automorphic representations ρ = ρp ⊗ ρpf ⊗ ρ∞ of GL2/Q such that:

• ρ∞ is the discrete series representation whose infinitessimal character has Harish-

Chandra parameter (12 ,32 − k) ∈ C2 = X∗(T )Cwhere we identify X∗(T ) with Z2 by

(n,m) :

a 0

0 b

7→ anbm.

• ρp is either a principal series or special representation ρ(ψ1, ψ2) with (say) ψ1 unram-

ified and ψ1(p) a p-adic unit.

Let ρ = ρp⊗ ρpf ⊗ ρ∞ denote the base change of ρ to GL2/K. Then ρ∞ is the irreducible

principal series corresponding to the character:

a ∗

0 b

7−→

(a

|a|

)n+1

|a|−n(b

|b|

)n+1

|b|−n

(To check this one switches to the corresponding representations of the Weil groups WR

and WC. ρ∞ corresponds to:

WR = C×⋊1, c −→ GL2(C)

by c 7−→

0 −1

1 0

C× ∋ z 7−→

z

12 z

32−k 0

0 z32−kz

12

and so ρ∞ is the representation corresponding to :

WC = C× −→ GL2(C)

z 7−→

(z|z|

)k−1|z|2−k 0

0(z|z|

)k−1|z|2−k

which corresponds to the Langlands’ quotient of the principal series described above, which

is in fact irreducible.)

126

Thus (eρUpp )× (ρpf )

Up → eH1, cusp(Γ1(Npr), Sk−2,k−2(C)) where:

Up =

a b

c d

∈ GL2(Op) | c ≡ 0 mod Npr , d ≡ 1 mod Npr

Up =

a b

c d

∈

∏

v 6 |pGL2(Ov) | c ≡ 0 mod N , d ≡ 1 mod N

Moreover ρUpp 6= 0 (resp. (ρpf )

Up 6= 0) implies that ρUpp 6= 0 (resp. (ρpf )

Up 6= 0). As p is split

for v | p ρv = ρp and evρv = eρp 6= 0.

In summary we have for each k ≥ 2 a cuspidal automorphic representation ρ = ρp ⊗ρpf ⊗ ρ∞ such that:

0 6= (eρUpp )× (ρpf )

Up → eH1, cusp(Γ1(Npr), Sk−2,k−2(C))

Thus by theorem 4.1 we are done.

2) This comes from using the Weil lifting.

More precisely let L/K be a quadratic extension which is Galois over Q and in which

π and π split. Let σ1, σ2, σ1, σ2 be the embeddings L → Kac, with the first two extending

the identity on K and with σi(α) = σi(α). Let∞1, ∞2 be the infinite places corresponding

to σ1 and σ2; and let v1, v2, v1, v2 be the primes above p corresponding to σ1, σ2, σ1, σ2.

We can find an integer M prime to p, all whose prime divisors split from K to L, and which

is such that, for each n ≥ 0, there are grossencharacters χ1, χ2 of l such that:

•χ1∞ : C× × C× −→ C×

(a, b) 7−→(a|a|

)n+1|a|−n

(b|b|

)n+1|b|−n

• conductor(χ1) divides M

• χ2 is a finite character

• conductor(χ2) divides M(v2)∞v1∞

• χ2 v2 × χ2 v1 : (Z×p )2 → C× equals χπ × χπ : (Z×p )

2 → C×

127

(This is possible because for ǫ in a subgroup of finite index in the units of L, σiǫ ∈ R>0 for

i = 1, 2 and so χ1∞(ǫ) = 1.) In this case p12χ1 v1(v1) and p

12χ1 v2(v2) are v1-adic units.

Then let ϑ denote the Weil lifting of χ1χ2 to an automorphic representation of GL2/K.

Then:

• ϑ is cuspidal as χ1∞1 6= χ1∞2

• ϑ∞ is the principal series representation corresponding to:

a ∗

0 b

7−→

(a

|a|

)n+1

|a|−n(b

|b|

)n+1

|b|−n

• ϑπ is the principal series representation ρ(φ1, φ2) with φ1 unramified, p12φ1(π) a v1-

adic unit and φ2|O×π= χπ|O×

π

• ϑπ is the principal series representation ρ(φ′1, φ′2) with φ′1 unramified, p

12φ′1(π) a v1-

adic unit and φ′2|O×π= χπ|O×

π

• (ϑpf )Up 6= 0 where Up is as above with N =M2

Thus:

0 6= (eϑUpp )⊗ (ϑpf )

Up → eH1(Γ1(Npr), Sn,n(C))

with Up also as above, for some r. As this is true for a single value of N and for all n ≥ 0

we are done by theorem 4.1.

4.7 Torsion

Lastly we shall use our results to exhibit torsion in the first homology group of certain

sheaves. First we shall recall briefly what is obvious. There are four things we might look

at which fall by Pontriagin duality into two pairs:

1. torsion in H1 with coefficients in Sn1,n2(O) ←→ lack of p-divisibility in H1 with

coefficients in Sn1,n2(K/O).

128

2. torsion in H1 with coefficients in Sn1,n2(O) ←→ lack of p-divisibility in H1 with

coefficients in Sn1,n2(K/O)

The obvious way to look for torsion is to consider the long exact sequence corresponding

to:

0 −→ Sn1,n2(O)α−→ Sn1,n2(O) −→ Sn1,n2(O/αO) −→ 0

In cohomology this gives for Γ a congruence subgroup:

0 −→ Sn1,n2(O/αO)Γ −→ H1(Γ, Sn1,n2(O))α −→ 0

which in some sense describes explicitly the torsion in this case. However in homology we

get:

H2(Γ, Sn1,n2(O)) −→ H2(Γ, Sn1,n2(O/αO)) −→ H1(Γ, Sn1,n2(O))α −→ 0

which seems to be very little help as H2 is as mysterious as H1. (Dually if we look at

0→ O/αO → K/O α→ K/O → 0 we again see that we get an answer for ii) but not for i).)

Our methods allow us to say something about case i) in the case n1 6= n2.

Theorem 4.2 Fix n1, n2 with n1 6= n2, and suppose

dim eH1, cusp(Γ1(Npr),Oπ) > 0, which is certainly the case if there is a Λ-adic eigenform

of level N . Then for t ≥ s ≥ r with n1 ≡ n2 ≡ 0 mod pt−s, H1(Γ1(Nps), Sn1,n2(Oπ)) and

H1, cusp(Γ1(Nps), Sn1,n2(Oπ)) have torsion of exponent at least pt.

Notes: 1) Examples of values of N and p for which we can apply the theorem are

provided easily by proposition 4.4 of the last section.

2) If H1, cusp(Γ1(Nps), Sn1,n2(Oπ)) has torsion of exponent of order pt then the same is

true for the relative homology H1(Γ1(Nps)\Z , ∂(Γ1(Nps)\Z), Sn1,n2(Oπ)) as we see from

the exact sequence:

. . . −→ H1(∂(Γ\Z), M ) −→ H1(Γ\Z, M ) −→ H1(Γ\Z , ∂(Γ\Z), M ) −→ . . .

3) These are not the most precise results that can be proved by these methods but give

a good indication of the type of question that can be treated.

129

4) We could deduce a less precise theorem from our general theory using commutative

algebra, but we shall go back to the method of proof as it yields more precise information

more easily.

Proof: Without loss of generality r = s ≥ 1.

We have from the proof of theorem 4.1 an isomorphism:

j∗ : eH1(Γ1(Np

t), Sn1,n2(O/πtO))∼−→ eH1(Γ1(Np

t),O/πtO)

As n1 ≡ n2 ≡ 0 mod pt−r we have that xn1xn2 ≡ 1 mod πt

∀x ∈ Gt,r = (1 + πrO)/(1 + πtO) then this map is Gt,requivariant. In this case propo-

sition 4.2 tells us that:

eH1(Γ1(Npr), Sn1,n2(O/πtO))

∼−→ eH1(Γ1(Npr),O/πtO)

The long exact sequence corresponding to:

0→ Sn1,n2(O/πtO)→ Sn1,n2(Kπ/Oπ) πt→ Sn1,n2(Kπ/Oπ)→ 0

and the fact that eSn1,n2(Kπ/Oπ)Γ1(Npr) = 0 (see the last part of proposition 4.2) imply

that:

eH1(Γ1(Npr), Sn1,n2(O/πtO)) ∼= eH1(Γ1(Np

r), Sn1,n2(Kπ/Oπ))πt

and hence that:

eH1(Γ1(Npr), Sn1,n2(Kπ/Oπ))πt ∼= eH1(Γ1(Np

r),Kπ/Oπ)πt

or dualising:

eH1(Γ1(Npr), Sn1,n2(Oπ))⊗O/πtO ∼= eH1(Γ1(Np

r),Oπ)⊗O/πtO

The theorem now follows at once, recalling that

dim eH1Eis(Γ1(Npr), Sn1,n2(Kπ)) = dim eH1Eis(Γ1(Np

r),Kπ). The same arguments ap-

ply to the cuspidal parts.

We can somewhat extend these results on torsion to the case in which p does not divide

the level. The crucial result will be:

130

Proposition 4.5 Let N be prime to p; n1, n2, n positive rational integers with n1, n2 > n;

n1 + n2 ≥ 2n + r; n1 ≡ n2 ≡ n mod pr. Then we have a map of Γ1(Npr)-modules:

j : Sn1,n2(O/πrO) −→−→ Sn,n(O/πrO)

such that if I = [Γ1(N)

pr 0

0 1

Γ1(Np

r)]j−1∗

then j∗ I = Tpr and so:

I : eH•(Γ1(Npr), Sn,n(O/πrO)) → H•(Γ1(N), Sn1,n2(O/πrO))

This map preserves the cuspidal parts.

Proof:This follows from lemma 1.1 of section 1.1 because if g =

pr 0

0 1

then

Γ1(N)gΓ1(Npr) = Γ1(Np

r)gΓ1(Npr) ∐

∐Γ1(Np

r)giΓ1(Npr) where each gi is of the form

∗ ∗0 p∗

(see Shimura [Sh2]). The fact that this preserves the cuspidal part can easilly

be checked as in section 4.2. From this we deduce:

Corollary 4.5 Under the same assumptions as the proposition we have:

eH•(Γ1(Npr), Sn,n(O/πrO)) → H•(Γ1(N), Sn1,n2(Kπ/Oπ))

This map preserves cuspidal parts.

Proof:This follows from the long exact sequence corresponding to:

0→ Sn1,n2(O/πrO)→ Sn1,n2(Kπ/Oπ) πr

→ Sn1,n2(Kπ/Oπ)→ 0

and the fact that:

Sn1,n2(Kπ/Oπ)Γ1(N) = Sn1,n2(Kπ/Oπ)SL2(O) = 0

Corollary 4.6 Let N be prime to p and n1 6= n2, n positive rational integers with n1, n2 >

n; n1 + n2 ≥ 2n + r; n1 ≡ n2 ≡ n mod pr. If dim eH1cusp(Γ1(Np

r), Sn,n(C)) > 0 then

H1, cusp(Γ1(N), Sn1,n2(Oπ)) and H1(Γ1(N), Sn1,n2(Oπ))

have torsion of exponent divisible by πr.

131

Proof:This follows as in the proof of theorem 4.2 using the fact that

dim eGn,n(Γ1(Npr),C) ≥ dimGn1,n2(Γ1(N),C) which was noted at the end of section 4.2.

For example taking n = 10 we find:

Example 4.1 Assume that p splits in K and p 6 |τ(p) where τ is Ramanujam’s function ( i.e.

p is ordinary for ∆(z) =∑τ(n)e2nπiz, the cuspidal elliptic modular function of weight 12

for SL2(Z)), and if n1 6= n2; n1, n2 > 10; n1+n2 > 20+ r; and n1 ≡ n2 ≡ 10 mod pr(p−1)

then

H1(SL2(O), Sn1,n2) and H1 cusp(SL2(O), Sn1,n2)

have torsion of exponent divisible by pr.

132

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