Images of Adelic Representations of Modular Forms

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MSc Mathematics

Master Thesis

Images of Adelic Representations of

Modular Forms

Author: Supervisor:

Robert Mann

dr. A.L. Kret

Examination date:

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Abstract

Serre was able to prove in [18] that the image of adelic representation of elliptic curves without complex multiplication are open. In this thesis we explain and give proofs of the main results of David Loeffler’s paper Images of adelic Galois representations for modular forms[8], wherein he gives an analogue of Serre’s results for modular forms without complex multiplication. He is then able to generalise this result to finite products of representations of modular forms without complex multiplication, subject to certain conditions.

Title: Images of Adelic Representations of Modular Forms Author: Robert Mann, mann.robertj@gmail.com, 10524940 Supervisor: dr. A.L. Kret

Second Examiner: prof. dr. L.D.J. Taelman Examination date: August 7, 2018

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

In loose terms1, modular forms of weight k and level N are holomorphic functions on the upper-half plane that have symmetry properties relating to an integer k ≥ 2 and the congruence subgroup Γ1(N ) =    a b c d ! ∈ SL₂(Z) : a ≡ d ≡ 1 (mod N ) c ≡ 0 (mod N )    .

As such, they are analytic objects, but they also heavily connected to number theory via the Langlands program.

Crucially the set of modular forms for a chosen pair k ≥ 2, N of integers forms a finite dimensional vector space Mk(N ) over C. This vector space can be decomposed

into the space of modular forms that arise from a smaller choice of N , and the space of newforms. A chosen newform f (that satisfies some extra conditions) will give rise to a number field L, and the adelic Galois representation of f ,

ρf : GQ → GL2(L ⊗QA∞Q),

where GQis the absolute Galois group of Q, and A∞Q denotes the finite ad`ele ring of Q.

These representations are constructed from the Tate modules of abelian varieties Q whose sets of C-rational points are of the form Cg/Z2g—note that these varieties do depend on the choice of embedding Z2g ,→ Cg—and so they can be viewed as higher dimensional analogues of those constructed from elliptic curves where g = 1. In 1995 Andrew Wiles was able to prove that the representations of semi-stable elliptic curves are given by modular forms with k = 2, and in doing so he was able to prove Fermat’s last theorem. Later in 2001 it was proven that the same result holds for all elliptic curves, and so representations of modular forms can be seen as a generalisation of those for elliptic curves.

However the representations of modular forms are in some ways less well-behaved than those of elliptic curves. In particular ρf typically does not have open image,

whereas Serre proved in 1972 that representations of elliptic curves without complex multiplication always have open image.

In [10], Momose constructed an open subgroup H of GQ, a subextension F of L/Q,

and a quaternion algebra B over F such that for each prime number `, the `-adic representation ρf,`|H has open image inside the subgroup of elements (B ⊗QQ`)× with

norm inside Q×(k−1)_` , where Q×(k−1)_` denotes the group of (k − 1)-th powers inside Q×_`.

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Ribet then built on this result in [16], demonstrating that for all but finitely many primes `, B ⊗QQ` = M2(F ⊗QQ`), and the image of ρf,`|H is conjugate to the subgroup of

GL2(OF ⊗ZZ` with determinant a (k − 1)-th power in Z×` . After seeing these results

one would expect that the image of ρf|H inside {x ∈ (B ⊗QA∞Q)× : det x ∈ Z ×(k−1)

` }

would also be open. However this result does not follow immediately for a fairly simple group-theoretic reason: the group ρf(H) does not necessarily contain ρf,`(H), where

ρf,`(H) is identified withQp6=`{1} × ρf,`(H) inside (B ⊗QA∞Q).

Loeffler was able to demonstrate in [8] that ρf|H × χ—where χ is the cyclotomic

character—has open image in the group {(x, λ) ∈ (B ⊗QA∞Q)

×_{× A}∞

Q : normB/F = λk−1}.

Now let {f1, . . . , fn} be a finite set of modular forms, and denote the weight of fi

by ki. Denote the quaternion algebra corresponding to fi by Bi. Assume also that the

fi are pairwise “non-conjugate” (see the section “Inner Twists”). Then there exists an

open subgroup H of GQ, such that the image of (ρf1× · · · × ρfn)|H × χ is open inside

   (x1, . . . , xn, λ) ∈ n Y i=1 (Bi⊗QA∞Q)×× A ∞× Q : det xi = λ ki−1    .

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1 Preliminaries

We begin by defining modular forms and their adelic Galois representations. Also, as a precursor to the main topic of this thesis we will discuss Galois representations of elliptic curves. The reader who is familiar with these concepts can safely skip this chapter.

1.1 The absolute Galois group

The Galois representations we are interested in will always be defined on the Galois group GQ = Gal(Q/Q), and so the purpose of this chapter is to give this group the

structure of a topological group, to discuss how Galois theory applies to GQ, and to

explain what Galois representations on GQ are.

Let L/F be a Galois field extension. The case where L/F is of infinite degree is more complicated than the finite degree case—for instance subgroups of Gal(L/F ) do not correspond to intermediate fields. To have a Galois theory of infinite extensions we first need to give Gal(L/F ) the Krull topology, which is the group topology with basis sets around the identity of the form Gal(L/K), where K/F is a field extension of finite degree. Equivalently, we may describe Gal(L/F ) and the Krull topology as the profinite group given by the projective limit,

lim ←−

K

Gal(K/F ),

with the usual profinite topology, where K ranges over the finite Galois sub-extensions of L/F . The intermediate fields of L/F then correspond to closed subgroups of Gal(L/F ), and Galois sub-extensions to normal closed subgroups, in the usual inclusion-reversing manner.

If K is a field with separable field closure Ksep_{, then the absolute Galois group of K}

is GK = Gal(Ksep/K). Given an algebraic field extension F/Q, that is possibly infinite,

we write OF for the integer ring of F , and in the case F = Q we simply write O = OQ.

If p is a maximal ideal of OF then we write k(p) = OF/p for the residue field of p. A

finite place of F is a p-adic norm, where p is a maximal ideals of OF, and the infinite

places of F are the field norms given by embeddings F ,→ C. If an infinite place of F is given by an embedding F ,→ R, then we call that place a real place of F , and otherwise we call it a complex place of F . Note that complex places of F are always given by pairs of complex conjugate embeddings F ,→ C. We also refer to finite places of F as finite primes of F , and infinite places of F as infinite primes of F , and so on. If L/F is an algebraic field extension with w a place of L and v a place of F , then w is said to lie above v if a field norm corresponding to w can be restricted to a field norm corresponding to v. If w is a finite place then this means w is given by a finite prime P

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of L such that P ∩ F = p is the finite prime of F corresponding to v. When we refer to finite primes of F we will usually write p, P, or λ, unless if F = Q in which case we write p or `. When we refer to places (finite or infinite) of F we will use v or w. If v is the finite place corresponding to a prime p then we use Fv and Fp interchangeably, as

well as k(v) and k(p).

Assume now that F is a number field and K/F is a (possibly infinite) Galois extension. Let P be a finite prime of K, p ⊆ P a finite prime of F , and p ∈ p a prime number. Then k(p) ∼= Fqfor some prime power q = pn, and k(P)/k(p) is a Galois extension. The

decomposition group DP of P is the subgroup of Gal(K/F ) fixing P. Now, because DP

fixes P and F , it acts on k(P) in a way that fixes k(p), hence we have a surjection DP→ Gal(k(P)/k(p)),

whose kernel IP is called the inertia group of P. The group Gal(k(P)/k(p)) is either

finite cyclic or isomorphic to the profinite integers bZ. In either case it is topologically generated by the element σq : x 7→ xq, called the Frobenius element of Gal(k(P)/k(p)).

Any element of DP mapping to σq is called a Frobenius element of P, and is denoted by

FrobP. Assume now that p is unramified in K, which means that the inertia group of P

is trivial, then the Frobenius element FrobP is unique. Moreover if σ ∈ Gal(K/F ) then

σP is also a finite prime lying over p, and it follows that FrobσP= σ FrobPσ−1. In this

case the Frobenius elements of finite primes lying over p form a conjugacy class, so we may define Frobp to be the Frobenius element of some finite prime P lying over p, and

then Frobp is defined up to conjugacy.

We will now see that the Frobenius elements in GQ are dense; in fact we will see that

a stronger statement is true.

Theorem 1.1.1 (Chebotarev’s density theorem). Let K/F be a finite Galois extension of number fields of degree n, and let X be a union of conjugacy classes in Gal(K/F ), then the set of finite primes p of F whose Frobenius elements lie in X has density #X/n. Corollary 1.1.1.1. Let P be a set containing all but finitely many prime numbers, and for each p ∈ P and each finite prime p of Q lying over p pick a Frobenius element Frobp,

then these Frobenius elements are dense in GQ.

Definition 1.1.2. Let ` be a prime number. An `-adic representation of Gal(K/F ) is a continuous group homomorphism

ρ : Gal(K/F ) → GLn(Q`)

for some positive integer n. The representation ρ is called unramified at a finite prime p of F if ρ is trivial on the inertia groups over p.

Because Gal(K/F ) is a compact group, we can show that ρ is isomorphic over Q` to

a representation taking values in GLn(Z`). First, note that each `-adic lattice in Qn`

is conjugate to Zn_`, and so is fixed by a subgroup of GLn(Q`) conjugate to GLn(Z`),

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of ρ there exists a normal open subgroup H of Gal(K/F ) such that H fixes Λ. The subgroup H has finite index, so there exists a finite set {σ1, . . . , σj} of representatives

of Gal(K/F )/H, and the set Λ0 =Sj

i=1ρ(σi)Λ is a new `-adic lattice inside Qn` that is

fixed by Gal(K/F ). Hence the image of ρ is conjugate to a subgroup of GLn(Z`).

For each prime number p, let Frobp ∈ GQ be a Frobenius element of a finite prime

over p. Let ρ and τ be two semi-simple `-adic representations of GQ, and assume there

exists N ≥ 1 such that ρ and τ are unramified at all primes p ≥ N . If Tr(ρ(Frobp)) = Tr(τ (Frobp))

for all primes p ≥ N , then by corollary 1.1.1.1, the representations ρ and τ have the same character, and hence are conjugate as representations over Q`.

1.2 Elliptic Curves

In this section we discuss elliptic curves over Q, their Galois representations, and give the results proven by Serre in his work during the late sixties and early seventies. The very short version is that elliptic curves over Q can be seen as 2-dimensional tori C/Z2, and we can use these tori to construct Galois representations in a natural way. Later we will see that representations of modular forms are constructed from higher dimensional tori Cn/Z2n (n ≥ 1) in a similar way, and so in this way the representations of modular forms can be seen as a higher dimensional analogue of the representations of elliptic curves. And in fact due to the modularity theorem they are a generalisation.

An elliptic curve E over a field k is a non-singular projective k-variety defined by a cubic equation of the form

E : Y2Z + a1XY Z + a3Y Z2= X3+ a2X2Z + a4XZ2+ a6Z3

with a1, . . . , a6∈ k. This equation is called a Weierstrass equation for E. If the

charac-teristic of k is neither 2 nor 3 then by a change of variables we may write Y2Z = X3+ aXZ2+ bZ3

with a, b ∈ k, such that the discriminant −16(4a3 + 27b2) is non-zero. Note that the point at infinity, O = (0 : 1 : 0), is always a point of inflection.

Crucially we can give E the structure of an abelian variety over k. Let K be some field extension of k, and let P and Q be two distinct points on E(K), the line through P and Q intersects E(K) in a third point, denoted P Q. We then define P + Q = O(P Q). When P = Q we simply use the tangent line, and take P P to be the third point where the tangent line intersects E. Finally we have O + O = O. Because these operations are all defined by polynomials over k, we find that + gives E the structure of an abelian variety over k. For a more detailed argument the reader should consult [9, I.3]

Assume now that E is defined over C. We can use the Weierstrass ℘-function to find a biholomorphic group isomorphism E(C) → C/Λ, with Λ a full lattice in C. Given

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τ ∈ h, with h the upper-half complex plane, we write Λ = Z + τ Z, ℘(z; τ ) = 1 z2 + X w∈Λ\{0} 1 (z + w)2 − 1 w2 , g2 = 60 X w∈Λ\{0} w−4, and g3 = 140 X w∈Λ\{0} w−6.

One can then show by comparing the poles at 0 that

℘0(z; τ )2 = 4℘(z; τ )3− g2℘(z; τ ) − g3,

where τ is treated as a constant, and the differentiation is with respect to z. We then find that the function

C → C × C given by

z 7→ (℘(z; τ ), [℘0(z; τ )/2]2)

gives a biholomorphic group isomorphism between C/Λ and the C-rational points of the elliptic curve E : Y2Z = X3−g2 4 XZ 2₋g3 4 Z 3_.

In this way we find that E(C) is always a torus.

For the remainder of this section we assume that E is defined over Q. Now, E can also be studied through its torsion groups, defined by

E[N ] =    x ∈ E(C) : N · x = x + · · · + x | {z } N = O    , N ∈ Z≥1.

Because the group structure is given by polynomials over Q, the torsion points are all Q-rational points. Taking all the torsion points over all N we get a dense subset of E, and so by taking the projective limit of the torsion groups we may study all these points simultaneously. We call this projective limit the Tate module, and denote it by T (E). Additionally, using the isomorphism E(C) ∼= C/Z2, we find that

E[N ] ∼= Z/N Z × Z/N Z, which gives

T (E) ∼= bZ × bZ.

Finally we define an action of GQ on E(Q) via σ(x : y : z) = (xσ, yσ, zσ). Because

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addition, and so it acts on each torsion group E(N ), and hence the Tate module T (E). The result is that we have a representation

ρE : GQ→ GL2(bZ)

that is well-defined up to conjugacy.

To see that ρE is continuous, consider U = 1 + N M2(bZ). The preimage of U under

ρE is simply Gal(Q/Q(E[N ])), which is open.

The image of ρE depends largely on the structure of the ring R of Q-endomorphisms

of E, or to be more precise if the natural inclusion Z ,→ R is an isomorphism or not. If we have Z 6∼= R then we say that E has complex multiplication. To see what other endomorphisms are possible we consider endomorphisms of E(C) viewed as a torus. Any non-trivial endomorphism of E(C) is given by multiplication by some non-zero λ ∈ C satisfying λΛ ⊆ Λ. Let R be the ring of endomorphisms of E. First, each member of Z gives an endomorphism of E, so Z ⊆ R. If there exists α /∈ R \ Z, then by considering the action of α on ω1 and ω2 we find that ω1/ω2 satisfies a quadratic equation over Z,

and so Q(ω1/ω2) is an imaginary quadratic field, and so R is an order inside Q(ω1/ω2).

Finally we come to Serre’s result for Galois representations of elliptic curves:

Theorem 1.2.1. Let E be an elliptic curve over Q without complex multiplication, then the image of ρE in GL2(bZ) is open.

Proof. See [18, theorem 3].

For each prime number ` we naturally obtain a representation ρE,`given by projection

from GL2(bZ) to GL2(Z`), and we can immediately see that when E is without complex

multiplication, the representation ρE,` always has open image, and is surjective for all

but finitely many prime numbers `.

1.3 Modular Forms

Modular forms are particularly well-behaved holomorphic functions define on the upper-half plane which satisfy symmetry properties relating to SL2(Z). These modular forms

are linked to number theory in the sense that many functions of interest to number theorists, such as the partition function and the divisor sum function, can be used to construct modular forms, and then those modular forms can in turn be used to prove properties about the original number theoretic object. They can also be used to construct Galois representations which are of interest to number theorists because they give us a way to study GQ. Define an action of SL2(Z) on P1(C) by γ · (z : w) = (az + bw : cz + dw), γ = a b c d ! ∈ SL₂(Z), (z : w) ∈ P1(C).

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We write ∞ = (1 : 0) and identify C with a subspace of P1(C) via the inclusion map C ,→ P1(C); z 7→ (z : 1).

Then, by an abuse of notation, we write γ · z = az + b

cz + d, γ ∈ SL2(Z), z ∈ P

1_(C).

with the understanding that if z = ∞, then γ · z = a/c, and if cz + d = 0 then γ · z = ∞. By computing the action of SL2(Z) on the imaginary part of a complex number we

find that the action of SL2(Z) on C restricts to an action on the upper-half plane

h= {z ∈ C : =z > 0}, and we also find that it gives an action on P1(Q); we will use these two actions in particular in the definition of a modular form. We also write

j(γ, z) = cz + d and

(f |kγ)(z) = j(γ, z)−kf (γ · z)

where f is a complex-valued function on the upper-half plane h, and k is any integer. We may check that this gives a group action on the set of holomorphic functions h → C. Definition 1.3.1 (Congruence subgroups). Given a positive integer N , we write

Γ(N ) :=γ ∈ SL2(Z) : γ ≡ 1 (mod N ) ,

where 1 denotes the identity matrix. We call Γ(N ) the principal congruence subgroup of level N . If Γ is a subgroup of SL2(Z) containing Γ(N ) for some N , then Γ is called

a congruence subgroup, and the level of Γ is the smallest positive integer N satisfying Γ(N ) ⊆ Γ.

Note that Γ(N ) is the kernel of the modulo N reduction map SL2(Z) → SL2(Z/N Z),

so Γ has finite index in SL2(Z). (The reduction map is surjective, but because we do

not need this fact we will not prove it.) For the remainder of this section N will denote a positive integer, Γ a congruence subgroup of level N , and k an arbitrary integer. Definition 1.3.2 (Weakly modular functions). A function f : h → C is weakly modular of weight k with respect to Γ if it is meromorphic, and for all γ ∈ Γ we have

f |kγ = f.

Definition 1.3.3 (Cusps). Two points x, y ∈ P1(Q) are called Γ-equivalent if there exists γ ∈ Γ such that γ · x = y. A maximal set of Γ-equivalent points in P1(Q) is called a cusp of Γ.

Equivalently, we can define a cusp c of Γ to be a member of Γ\ SL2(Z)/ SL2(Z)∞,

where SL2(Z)∞ is the subgroup of SL2(Z) fixing ∞. The two definitions can be seen

to be equivalent by saying that r ∈ P1(Q) and α ∈ SL2(Z)/ SL2(Z)∞ represent the

same cusp if and only if α · ∞ is Γ-equivalent to r. We denote the set of cusps of Γ by Cusps(Γ), and we will typically use both definitions of a cusp interchangeably.

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Let c be a cusp of Γ, let α ∈ SL2(Z) be a representative of c, and let hcbe the smallest

positive integer such that

1 hc

0 1 !

∈ α−1Γα ∩ SL2(Z)∞.

Let f be weakly modular of weight k with respect to Γ, then f |kα is periodic with period

hc, so by letting qc= exp(2πiz/hc), we may define a function df |kα : D \ {0} → C (where

D denotes the unit disk in the complex plane) that satisfies df |kα(qc) = f (z). Now df |kα

is holomorphic on its domain if and only if f |kα is holomorphic on h, and we say f |kα

is holomorphic at ∞, or that f is holomorphic at c, if f |kα extends holomorphically to

a function df |kα : D → C. When this occurs we have a Fourier series

f |kα(z) = ∞

X

n=0

an(f |kα)qnc.

Additionally if a0(f |kα) = 0, we say that f |kα vanishes at ∞, or that f vanishes at c.

Definition 1.3.4 (Modular forms). A function f : h → C is a modular form of weight k with respect to Γ if it is holomorphic, weakly modular of weight k with respect Γ, and if f is holomorphic at each cusp of Γ. The set of modular forms of weight k with respect to Γ is denoted by Mk(Γ), and forms a complex vector space of finite dimension.

Additionally, if f also vanishes at every cusp of Γ, then f is called a cusp form of weight k with respect to Γ. The set of cusp forms of weight k with respect to Γ is denoted by Sk(Γ), and forms a linear subspace of Mk(Γ).

Remark 1.3.4.1. The direct sum

M(Γ) =M

k∈Z

M_k(Γ),

with point-wise addition and multiplication, forms a graded ring with graded ideal S(Γ) =M

k∈Z

S_k(Γ).

In this paper we will frequently refer to the following congruence subgroups:

Γ0(N ) :=    a b c d ! ∈ SL₂(Z) : c ≡ 0 (mod N )    , and Γ1(N ) :=    a b c d ! ∈ Γ0(Z) : a ≡ d ≡ 1 (mod N )    .

A member of Mk(Γ1(N )) will be referred to as a modular form of level N and weight

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Definition 1.3.5 (Diamond operators). Observe that the map

Γ0(N ) → (Z/N Z)×; a b

c d !

7→ d

is a surjective homomorphism with kernel Γ1(N ). Therefore Γ1(N ) is a normal subgroup

of Γ0(N ) and we have Γ0(N )/Γ1(N ) ∼= (Z/N Z)×. It follows that we may define an

action of Γ0(N ) on Mk(Γ1(N )) by γ · f = f |kγ, and so we may use the congruence

Γ0(N )/Γ1(N ) ∼= (Z/N Z)× to define an action of (Z/N Z)× on Mk(Γ1(N )). We call

this action the diamond operator, and we denote it by hdif for d ∈ (Z/N Z)× and f ∈ Mk(Γ1(N )).

Definition 1.3.6 (Dirichlet characters). A group homomorphism (Z/N Z)× → C× _is

called a Dirichlet character. For each Dirichlet character χ there exists a minimal N0 | N and a Dirichlet character χ0: (Z/N0Z)×→ C× such that χ is the composition

(Z/N Z)×→ (Z/N0Z)× χ

0

−→ C×.

We call N0 the conductor of χ, and χ is called primitive if N = N0.

Definition 1.3.7 (Nebentypus). Let χ : (Z/N Z)× → C× be a Dirichlet character. A modular form f ∈ Mk(Γ1(N )) is said to have nebentypus χ if

hdif = χ(d)f

for all d ∈ (Z/N Z)×. Alternatively f is said to be of character χ. Note that f ∈ S_k(Γ0(N )) precisely when f has nebentypus 1.

Definition 1.3.8 (Hecke operators). For each prime number p we define an operator Tp on Mk(Γ1(N )) by

Tpf =

X

β

f |kβ,

where β ranges over a set of representatives of

Γ1(N )\Γ1(N )

1 0 0 p

!

Γ1(N ).

The diamond operators and the Tp operators form the Hecke operators.

Lemma 1.3.9. The diamond operators and the Tp operators generate a commutative

Z-algebra, denoted T(Mk(Γ1(N ))), and referred to as a Hecke algebra.

Proof. [3][Proposition 5.3.1.]

In our study of newforms it will be important to have an inner-product on Sk(Γ1(N )).

This will allow us to decide which cusp forms of level N come from cusp forms of a lower level, and which are canonically of level N .

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Definition 1.3.10 (Petersson inner product). Let

D = {z ∈ h : −1/2 ≤ <z ≤ 1/2 and |z| ≥ 1}.

Note that every point in h is either SL2(Z)-equivalent to two points in D, in which

case those two points lie on the boundary, or just one point in D. We call D the fundamental domain. Now letting R be a system of representatives for Γ\ SL2(Z) we

write DΓ =S_γ∈RγD.

The Petersson inner product h−, −iΓ on Sk(Γ) is defined by:

hf, giΓ=

Z

z∈DΓ

f (z)g(z)ykdx dy y2 ,

where x and y are the real and imaginary parts of z respectively.

Definition 1.3.11 (Oldforms and newforms). Let M be a positive integer dividing N , and let d divide N/M . We may embed Sk(Γ1(M )) into Sk(Γ1(N )) via f 7→ f |kαdwith

α = d 0 0 1

! .

For each d | N we let id be the map

id: (Sk(Γ1(N d−1)))2→ Sk(Γ1(N ))

defined by

(f, g) 7→ f + g|kαd.

The space of oldforms of level N is S_k(Γ1(N ))old=

X

p

ip((Sk(Γ1(N p−1)))2),

where p ranges over the prime divisors of N , and the space of newforms of level N is the orthogonal complement with respect to the Petersson inner product:

Sk(Γ1(N ))new = (Sk(Γ1)old)⊥.

Definition 1.3.12. An eigenform, sometimes called a Hecke eigenform is a non-zero modular form f ∈ Mk(Γ1(N )) that is an eigenvector for all Hecke operators Tp for

p prime, and all diamond operators hdi with d ∈ (Z/N Z)×. We call f a normalised eigenform if a1(f ) = 0. A primitive form of weight k for Γ1(N ) is a normalised eigenform

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1.4 Galois representations of Modular Forms

The purpose of this section is to give a very rough sketch of the construction of the Galois representations of a normalised cuspidal eigenform of weight 2, level N , and nebentypus ε. In this thesis we will be considering representations of newforms of weight k ≥ 2, however the construction of these representations is substantially more compli-cated than the weight 2 case, and falls outside the scope of this thesis. The construction we give here is due to Shimura, building upon the Eichler-Shimura relation. For a full treatment of the weight 2 case the reader is directed to [3], A First Course in Modular Forms by Fred Diamond and Jerry Shurman. For a treatment of the weight ≥ 2 case, see [5, chapter 2].

Unlike in our treatment of Galois representations of elliptic curves we only look at a single prime number `, and construct a representation

ρf,`: GQ → GL2(L ⊗QQ`),

where L is the number field attached to f .

Given a congruence subgroup Γ, one can show that the set Y (Γ)C = Γ\h with the

quotient topology is a non-compact Riemann surface, see for example chapter 2 of [3]. When showing that Y (Γ)Cis a Riemann surface one By adding points corresponding to

the cusps of Γ we obtain a compact Riemann surface X(Γ)C. For ease of notation, we

write

Y (Γ0(N ))C= Y0(N )C, Y (Γ1(N ))C= Y1(N )C, and Y (Γ(N ))C= Y (N )C,

as well as

X(Γ0(N ))C= X0(N )C, X(Γ1(N ))C= X1(N )C, and X(Γ(N ))C= X(N )C.

We can also consider X1(N )Cto be the set of C-rational points of a variety over Q, by

which we mean that there exists a projective nonsingular algebraic curve X1(N ) over Q,

such that X1(N )(C) with its analytic topology is isomorphic to X1(N )C, see [3, p.386].

We will mainly be interested in X1(N )(Q) and X1(N )C, so for ease of notation we refer

to the former set as X1(N ).

First we construct an action of GQ on the `-adic Tate module of the Jacobian group

J1(N ) = Jac(X1(N )C). To do this we let GQ act on the divisor group of X1(N ) in a

coordinate-wise manner, or written symbolically,

σP = Pσ for P ∈ X1(N ), σ ∈ GQ.

This action then generates an action on the Picard group Pic0(X1(N )), and so on

the `n_{-torsion group Pic}0_(X

1(N ))[`n]. This torsion group is in fact isomorphic to

Pic0(X1(N )C)[`n], see [3, p.386]. So using the isomorphism J1(N ) ∼= Pic(X1(N )C),

we have an action of GQ on T`(J1(N )).

On the other hand J1(N ) = Jac(X1(N )C) is known to be isomorphic to a torus Cg/Λ,

where Λ is a full-lattice in Cg. Therefore, we have an isomorphism, T`(J1(N )) = lim_←−

n≥1

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From this we obtain an action of GQ on the Tate module of J1(N ), and hence a

repre-sentation

ρX1(N ),`: GQ→ Aut(T`(J1(N ))) ∼= GL2g(Z`),

which is the 2g-dimensional `-adic representation attached to X1(N ). As with elliptic

curves this representation is only defined up to conjugation. Of course this representa-tion depends only on the level of f , so the next step is to use ρ_X₁_{(N ),`} to construct a representation, does depend on f .

We will now define an action of the Hecke algebra T = T(Mk(Γ1(N ))) for Γ1(N ) on

J1(N ). Recall that the Weil pairing eN of an elliptic curve E = C/(ω1Z + ω2Z) is the

function eN : E[N ] × E[N ] 7→ µN, defined by eN(P, Q) = exp(2πi det γ/N ), where γ ∈ M2(Z/N Z) satisfies P Q ! = γ ω1/N ω2/N ! inside E[N ].

Definition 1.4.1 (Enhanced elliptic curves). An enhanced elliptic curve for Γ ∈ {Γ0(N ), Γ1(N ), Γ(N )}

is an ordered pair (E, X), where E is an elliptic curve over C, and X is a point in E with order N if Γ = Γ1(N ), a cyclic subgroup of E with order N if Γ = Γ0(N ), or a pair

(P, Q) ∈ E × E with hP, Qi = E[N ] and eN(P, Q) = e2πi/N if Γ = Γ(N ).

Two enhanced elliptic curves (E, X) and (E0, X0) for Γ are equivalent, denoted (E, X) ∼ (E0, X0), if there exists an isomorphism of elliptic curves ϕ : E −→ E∼ 0 with ϕ(X) = X0. If X is an ordered pair then ϕ acts on X entry-wise.

The equivalence classes of enhanced elliptic curves for Γ0(N ), Γ1(N ), and Γ(N ) are

denoted S0(N ), S1(N ), and S(N ) respectively.

It turns out that we can naturally identify S0(N ), S1(N ), and S(N ) with the Riemann

surfaces Y0(N ))C, Y1(N )C, and Y (N )C, respectively. First note that every member of

S0(N ) is represented by an enhanced elliptic curve of the form (Eτ, h1/N + Λti) where

τ ∈ h, Λτ = Z + τ Z, and Eτ = C/Λτ. Furthermore we have

(Eτ, h1/N + Λτi) ∼ (Eτ0, h1/N + Λ_τ0i)

precisely when τ = γτ for some γ ∈ Γ0(N ), so S0(N ) bijects naturally with Y0(N )C.

And similarly we can identify S1(N ) with Y1(N ), and S(N ) with Y (N ).

Now T acts on the group of divisors for S1(N ) by

Tp[E, Q] =

X

C

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where C varies over the order p subgroups of E, and hdi[E, Q] = [E, dQ].

We can also use this action to define an action of T on the Picard group of X1(N )C.

We begin by defining an action of T on Div(X1(N )). We have Tp : X1(N ) → Div(X1(N ))

defined by,

Γ1(N )τ 7→

X

β

Γ1(N )β,

where β ranges over a set of representatives of,

Γ1(N )\Γ1(N )

1 0 0 p

!

Γ1(N ),

and hdi : X1(N ) → X1(N ) defined by,

Γ1(N )τ 7→ Γ1(N )ατ,

where α is a member of Γ0(N ) with bottom-right entry congruent to d modulo N . Both

of these maps naturally extend to maps Div(X1(N )) → Div(X1(N )), in a way that

commutes with the inclusion S1(N ) ∼

−→ Y1(N ) ,→ X1(N ) given by

[Eτ, 1/N + Λτ] 7→ Γ1(N )τ.

A less obvious—but nonetheless true—fact is that this action can be restricted to the Picard group of X1(N ), so that the diagrams

Div0(S1(N )) Div0(S1(N )) Pic0(X1(N )) Pic0(X1(N )) Tp Tp and Div0(S1(N )) Div0(S1(N )) Pic0(X1(N )) Pic0(X1(N )) hdi hdi commute.

Finally because Pic0(X1(N )C) is isomorphic to the Jacobian J1(N ) we have an action

of T on J1(N ). To f we associate an ideal If of T defined by

If = {T ∈ T : T f = 0},

and an abelian variety

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Now we have an isomorphism

T−→ O∼ f, where Of = Z[an(f ) : n ≥ 1],

given by

Tp 7→ ap(f ) and hdi 7→ ε(d)

We then find that the fraction field of Of is the number field L attached to f , and

using the above isomorphism we obtain an action of Of on Af. Moreover if d = [L : Q],

then Af is a torus of complex-dimension d, so the `-adic Tate module of Af is isomorphic

to Z2d_` . The action of T on Af defines an action of Of on T`(Af).

For each prime power `n we have a natural surjective map of `n-torsion groups Pic0(X1(N ))[`n] → Af[`n],

which is surjective with kernel stable under GQ. So GQ acts on T`(Af) also. And this

action commutes with that of T.

Putting all this together gives a Galois representation ρAf,`: GQ→ GL2d(bZ).

Because T`(Af) is an Of-module, the tensor product T`(Af) ⊗Z Q is an L ⊗QQ`

-module, and moreover it is free of rank 2, see [3, lemma 9.5.3], so we have a representation ρf,`: GQ → GL2(L ⊗QQ`).

An important result in the development of representations of modular forms is the Eichler-Shimura relation. For this we need a way to reduce the modular curve X1(N )

modulo p, where p - N is a prime number. To do this we use the moduli interpretation Y1(N )C ∼= S1(N ). This gives us that Y1(N ) is a scheme over Z[1/N ]. To be more

accurate, let R be a Z[1/N ]-algebra, and F (R) be the set of isomorphism classes of pairs (E, P ) where E is an elliptic curve over R and P ∈ E(R) is a point of order N . This functor is representable by a Z[1/N ]-scheme Y1(N ) (here we do not mean the Q-rational

points inside Y1(N )C). Extending our scheme to include the cusps of Y1(N ) we obtain

the scheme X1(N ) over Z[1/N ], which is an integral model of X1(N )C. We then write

e

X1(N ) = X1(N ) ×Spec Z[1/N ]Spec Fp.

Theorem 1.4.2 (Eichler-Shimura relation). Let p - N . The Hecke operator Tp acts on

Pic0( eX1(N )) by

Tp = σp+ σtp,

where σp denotes the Frobenius map in GFp and σ

t

p is the dual of σp.

Proof. For a proof of this theorem see chapter 8 of [3]; the theorem itself is given at the end of the chapter as theorem 8.7.2.

This relation leads to the following corollary describing the action of the Frobenius elements.

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Corollary 1.4.2.1. Let p - `N , then the action of Frobp on (L ⊗QQ`)2 satisfies

Frob2_p−T_pFrobp+p = 0.

Proof. By [3, theorem 9.5.1], the action of Frobpon T`(Pic(X1(N )) satisfies the equation

above. Hence it satisfies that same equation on Af, and hence also on (L ⊗QQ`)×.

Finally we are able to describe the Galois representation attached to f in the following theorem.

Theorem 1.4.3 (Eichler, Shimura). Let f ∈ S2(N, χ)newbe a normalised eigenform with

number field L, and let ` be a prime number, then there exists a Galois representation ρf,`: GQ → GL2(L ⊗QQ`),

such that if p is a prime number not dividing `N , then ρf,` is unramified at p, and

ρf,`(Frobp) is a root of

x2− ap(f )x + χ(p)p = 0.

Moreover, ρf,` is unique up to conjugation.

Proof. See [3, theorem 9.5.4].

Constructing the representations of eigenforms with weight greater than 2 is substan-tially more complicated, and involves the use of ´etale cohomology to construct a space upon which GQ acts in an appropriate way. Therefore we will simply state the result,

which is very similar to the weight 2 case.

Theorem 1.4.4 (Deligne [2]). Let k ≥ 2, f ∈ Sk(N, χ)new be a normalised eigenform

with number field L, and let ` be a prime number, then there exists a Galois representa-tion

ρf,`: GQ → GL2(L ⊗QQ`),

such that if p is a prime number not dividing `N , then ρf,` is unramified at p, and

ρf,`(Frobp) is a root of

x2− ap(f )x + χ(p)pk−1= 0.

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2 Why study Galois representations of

modular forms and elliptic curves?

One of the major insights of the 20th century was the Taniyama–Shimura–Weil conjec-ture, which stated in loose terms that the representations of elliptic curves over Q are also representations of eigenforms of weight 2. This conjecture started receiving a lot of attention in the eighties when Frey and Ribet demonstrated that if proven true, the conjecture would prove Fermat’s last theorem, and it was by proving that the conjecture holds for a certain class of elliptic curves that Wiles was able to prove Fermat’s last theorem. In this chapter we will briefly explain why Wiles’ work was sufficient to prove Fermat’s last theorem.

2.1 The modularity theorem

Let E be an elliptic curve over Q, then we may assume without loss of generality that E is given by

E : y2z + a1xyz + a3yz2 = x3+ a2x2z + a4xz2+ a6z3, a1, . . . , a6∈ Z.

This equation is called a Weierstrass equation, and has the advantage of being well-defined over all commutative rings, and so in particular we may consider the variety eE over Fpgiven by reducing the equation of E modulo p (in this section p and ` will always

denote prime numbers). We say that E has good reduction at p if eE does not have any singularities, and that E has bad reduction at p otherwise, in which case the singularity of eE lies in eE(Fp) and is unique. The only primes where E may have bad reduction are

those primes dividing the discriminant of E, so we may associate a positive integer N to E which encodes the type of reduction E has at each prime number, that is to say p | N if and only if E has bad reduction at p, and if p | N then the exponent of p in N encodes the type of singularity E(Fp) has. We call N the conductor of E.

In the fifties and sixties Yutaka Taniyama, Goro Shimura, and Andr´e Weil came up with the Taniyama–Shimura–Weil conjecture—which has since been proven and is now called the modularity theorem—which suggested that all elliptic curves over Q are modular. There exist many equivalent definitions of what it means for an elliptic curve to be modular, but the definition most relevant to this thesis states that E is modular if there exists f ∈ S2(Γ0(N )), a finite prime λ of L = Q(an(f ) : n ≥ 1), and a prime

number ` such that Q`∼= Lλ, and their `-adic and λ-adic representations are isomorphic

over Q`, i.e.

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In this case we also say that E is modular by f with respect to `. To see why this is nice we define a new quantity:

ap(E) = p + 1 − #E(Fp).

It then turns out that

Tr ρE,`(Frobp) = ap(E),

and so the modularity theorem tells us that there exists f ∈ S2(Γ0(N )) satisfying

ap(E) = ap(f ) whenever p - `N —to see this compare the characters of ρf,λ and ρE,`

when E is modular by f with respect to `. This version of the modularity theorem further implies that ap(E) = ap(f ) for all p; instead of proving this we direct the reader

to page 392 of [3]. Now, because the trace is a continuous map from GL2(Q`) to Q`,

and because the Frobenius elements Frobp with p - `N are dense (recall Chebotarev’s

theorem), we can now go a step further and say that E is modular by some f with respect to all `.

To conclude this section we briefly give an example of a use for this theorem. Consider the elliptic curve

E : y2z + yz2 = x3− x2z.

This curve has conductor 11, and S2(Γ0(N )) = Cf , where f is the eigenform

f = qY

n≥1

(1 − qn)2(1 − q11n)2.

So according to the modularity theorem we have that ap(E) = ap(f ) for all p. Also, the

elliptic curves

E0: y2z + y = x3− x2_{z − 10xz}2_{− 20z}3

and

E00: y2z + yz2 = x3− x2z − 7820xz2− 263580z3

also have conductor 11, and so because f is the only candidate eigenform for elliptic curves of conductor 11 to be modular by, we find that

ap(E) = ap(E0) = ap(E00)

for all primes p.

2.2 Fermat’s last theorem

The Taniyama–Shimura–Weil conjecture received a lot of attention in the eighties when it was shown that a example to Fermat’s last theorem would provide a counter-example to the conjecture. In the nineties Wiles was able to show

Theorem 2.2.1 (Fermat’s last theorem). Let ` ≥ 3 be an integer, then there do not exist integers a, b, c 6= 0 with

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We can immediately see that it suffices to consider the cases where ` is a prime number, gcd(a, b, c) = 1, b is even, and 4 | a + 1. Let us suppose that a, b, c, ` is a counter-example to Fermat’s last theorem. In 1984 Gerhard Frey noted that such a counter-example would give rise to an elliptic curve E defined by

E : y2z = x(x − a`z)(x + b`z).

The curve E is called either a Frey curve or a Frey–Hellegouarch elliptic curve. Assuming the Taniyama-Shimura-Weil conjecture to be true, E would then be modular. Frey believed that in fact E could not be modular, and so a counter-example to Fermat’s last theorem would disprove the Taniyama-Shimura-Weil conjecture. In 1987 [19] Serre then came up with the epsilon conjecture, and demonstrated that this conjecture would be sufficient to prove Frey correct, and in 1990 [12] Ribet was able to prove this conjecture, and so this conjecture is now called Ribet’s theorem. The upshot was that there would exist f ∈ S2(Γ0(2)) and a finite prime λ of Lλ = Q`, but instead of ρf,λ∼ ρE,`we would

have

¯

ρE,`∼ ¯ρf,λ: GQ→ GL2(F`),

where ¯ρE,` and ¯ρf,λ are the reductions of ρE,` and ρf,λ modulo ` and λ respectively.

However such an f cannot exist because the genus of X0(2) is 0, giving S2(Γ0(2)) = {0}.

Thus proving Fermat’s last theorem was reduced to proving that the Taniyama-Shimura-Weil conjecture holds for Frey curves, which Andrew Wiles and his student Richard Taylor were able to do in [23] and [22], when they proved that the conjecture holds for semi-stable elliptic curves—elliptic curves E over Q such that if E has bad reduction at p then E(Fp) has a double point—which includes the Frey curves. The full conjecture

was later proven by Breuil, Conrad, Diamond, and Taylor, and so is now called the modularity theorem.

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3 Images of adelic representations of

modular forms

The purpose of this chapter is to explain and prove the main results of the sections “Large image results for one modular form”, wherein Loeffler tries to find an analogue for Serre’s theorem that the representations of elliptic curves have open image, and “Joint large image” in Loeffler’s paper [8], where Loeffler discusses images of Cartesian products of Galois representations. First we pick an eigenform f , construct an open subgroup H of GQ, and reduce the codomain of the representations ρf of f such that

the restriction ρf|H can be said to be open. Then we find that if g is a second eigenform

with coefficients independent to those of f then the image of ρf× ρg|H can still be said

to be open in our new codomain.

Remark 3.0.0.1. We should mention that Loeffler’s original motivation for his paper is his fourth section, which discusses the existence of certain elements in the image of representations of the form ρf,p× ρg,p. In this thesis we will be focusing on the two

preceding sections.

3.1 Is the image open?

Recall theorem 1.2.1, which stated that if E is a elliptic curve without complex multi-plication, then the image of ρE in GL2(bZ) is open. On a purely intuitive level we might

expect this result to generalise to modular forms by simply saying that Galois represen-tations of modular forms also have open image in GL2(A∞_Q). However, we will now see

that this obvious generalisation does not work.

As before let f be a normalised cuspidal newform of weight k ≥ 2, level N , nebentypus ε, q-expansion P∞

n=1anqn, Fourier coefficient field L = Q(an : n ≥ 1), and with Galois

representation

ρf : GQ → GL2(L ⊗QA∞Q).

Let p be a prime of L lying over the prime number p, and let ` be a prime number not dividing pN . By theorem 1.4.4, we see that det ρf,p(Frob`) = ε(`)`k−1, so there exists

some finite power of det ρf,p that takes values in Z×p, which typically does not have finite

index in O_L,p× , in particular whenever Qp is a proper subfield of Lp.

Over the course of the next few sections we will formulate an analogue to Serre’s result for elliptic curves which will apply to modular forms without complex multiplication of weight k ≥ 2.

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3.2 Inner Twists

Let If denote the set of field inclusions L ,→ C. Because L is a number field it is natural

to ask, what happens if we apply some σ ∈ If to the q-coefficients of f ? The answer is

that we get another modular form,

fσ =

∞

X

n=1

aσ_nqn,

of the same weight and level. We call fσ a conjugate of f .

Alternatively let χ be a primitive Dirichlet character with conductor r, then there exists a unique newform f ⊗ χ with the property

ap(f ⊗ χ) = χ(p)ap(f )

for all but finitely many prime numbers p (see section 3 of [15]). The newform f ⊗ χ has the same weight as f , but if f ⊗ χ has level M then we can only say that N r and M r have the same prime factors.

If it happens that fσ _{= f ⊗ χ then (σ, χ) is called an inner twist of f . If there exists}

a non-trivial Dirichlet character χ such that f ⊗ χ = f , then f is said to have complex multiplication. From here on we will assume that f does not have complex multiplication. This means that given an inner twist (σ, χ), the character χ is uniquely determined by σ. Therefore, letting Γ be the set of inner-twists of f , we have an inclusion

Γ ,→ If,

and we write χ = χσ.

Remark 3.2.0.1. Saying that f has complex multiplication is equivalent to saying that there exists an imaginary quadratic field K such that ap = 0 whenever p is inert in K.

In this case the imaginary quadratic field K is unique, and as with elliptic curves with complex multiplication we have an intrinsic link between modular forms with complex multiplication and imaginary quadratic fields.

Next we have an important result about the set of inner-twists of f :

Lemma 3.2.1 (Momose [10, proposition 1.7]). If (σ, χ) is an inner-twist of f then σ(L) = L, and so Γ is a subset of Aut(L/Q). Moreover, if (γ, χ), (σ, µ) are inner twists of f , then (γ · σ, χσ _{· µ) is also an inner twist, and so Γ is a subgroup of Aut(L/Q).}

Furthermore Γ is abelian. Finally, writing F for the subfield of L fixed by Γ, the field extension L/F is a Galois extension, with Galois group Γ.

Proof. Here we write out the proof given by Momose. Let ˜ρf,` be the representation given by

˜

ρf,`: GQ ρf,`

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Now note that L may be viewed as a subfield of Q_`, and Q_` can be viewed as a subfield of C, and moreover each embedding σ ∈ If has image inside Q`. Therefore for each

σ ∈ If we let ˜σ be the composite

˜

σ : L ⊗QQ` → Q`

x ⊗ y 7→ σ(x)y. For each σ ∈ If, let ˜ρσf,` be the composite

˜ ρσ_f,` : GQ ˜ ρf,` −−→ GL₂(L ⊗QQ`) ˜ σ −→ GL₂(Q_`).

Let H be an open subgroup of GQ. By [14, 4.4] we know that ˜ρσf,`|H is irreducible for

each σ ∈ If. Moreover if H is normal, then ˜ρσ_f,`|H and ˜ρτ_f,`|H are not equivalent for all

σ 6= τ ∈ If, for otherwise there would exist a Dirichlet character χ 6= 1 such that ˜ρσf,` is

equivalent with ˜ρσ_f,`⊗ χ, making f a newform with complex multiplication.

Now, let (σ, χ) be an inner twist of f , so that ˜ρσ_f,` and ˜ρf,`⊗ χ are equivalent

repre-sentations. This gives

det ˜ρσ_f,` = χ2det ˜ρf,`,

or equivalently

χ2 = εσ/ε,

where χ and ε are viewed as characters on GQ in the usual manner. When ε has even

order we have εσ _{= ε}2j+1 _{for some integer j, giving χ}2 _{= ε}2j_{, and otherwise we have}

εσ = ε2j, so that χ2 = ε2j−1 = ε2j0 for some integer j0. In either case we have χ2 = ε2j for some integer j, thus χ = λ · εj with λ a character of order 1 or 2, and εσ = ε1+2i.

This proves that χ has image inside L, and hence so does σ, therefore given two inner-twists (σ, λεi) and (τ, µεj), we have

aστ_p = (µ · εj)(p)σa_pσ = (λ · µ · εi+j+2ij)(p)ap = aτ σp ,

for all but finitely many prime numbers p. The statement about composing inner-twists is now clear, thus Γ is an abelian subgroup of Aut(L/Q). To show that L/F is an abelian extension it suffices to show that Aut(L/F ) = Γ, so let σ ∈ Aut(L/Q) be such that σ|F = 1. It follows that

˜

ρσ_f,`|_{Gal(Q/F )}= ˜ρf,`|_{Gal(Q/F )}

and so there exists a Dirichlet character χ such that (σ, χ) is an inner-twist of f , hence σ ∈ Γ.

For n > 0 let µn denote the group of n-th roots of unity, and let µ∞denote the group

of all roots of unity. For each n there exists a group isomorphism Gal(Q(µn)/Q) ∼=

(Z/nZ)×. It follows that we may consider a Dirichlet character χ to be a character GQ→ Gal(Q(µn)/Q)

∼

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where n is the conductor of χ. Finally we take H to be

H = \

σ∈Γ

ker χσ ⊆ GQ.

Because H is the intersection of kernels of Dirichlet characters we can see that H is open in GQ, and because (complex conjugation, ε−1) is an inner-twist of f , the determinant

of ρf,`|H is simply χk−1, where χ is the cyclotomic character.

3.3 The quaternion algebra B

For the reader’s convenience let us restate the notation we are using. We take f = P

n≥1anqn to be a normalised cuspidal newform of weight k ≥ 2 on Γ1(N ) with

nebentypus ε. We assume that f does not have complex multiplication. We write L = Q(an: n ≥ 1) for the Fourier coefficient field of f , and we let Γ ⊆ Aut(L/Q) be the

group of inner twists of f . Each inner twists gives a Dirichlet character GQ → C×, and

we denote the intersection of the kernels of these characters by H. We recall also that L is an abelian extension of F with Galois group Γ. Also, if (γ, χ) ∈ Γ then we write χ = χγ, and we often refer to (γ, χγ) as just γ.

In the first subsection we will describe a quaternion algebra B over F such that ρf,`(H)

has open image inside (B ⊗QQ`)× for all prime numbers `, and full image for all but

finitely many `. The remainder of this chapter is then devoted to showing that ρf|H

has open image inside (B ⊗QA∞_Q)×. We will then have an analogue of Serre’s result

that images of adelic representations of elliptic curves without complex multiplication are open.

Let B be a quaternion algebra over F . For ease of notation we will now write G` = {x ∈ (B ⊗QR)×: normB/Fx ∈ Z ×(k−1) ` } and G = {x ∈ (B ⊗QA∞Q) ×_{: norm} B/Fx ∈ bZ×(k−1)}.

(Here by Z×(k−1)_` and bZ×(k−1), we mean the groups of (k − 1)-th powers.) Later we will be introducing algebraic groups G and G◦ over Q and Z respectively, so the use of G`

and G will only be temporary.

Theorem 3.3.1. [Momose, Ribet] There exists a quaternion algebra B over F with an embedding B ,→ M2(L) such that

ρf(H) ⊆ (B ⊗QA∞Q)×⊆ GL2(L ⊗QA∞Q).

Furthermore, the image ρf,`(H) is open in G` for all prime numbers `, and for all but

finitely many prime numbers ` we have B ⊗QQ` = M2(F ⊗QQ`), and rhof,`(H) =

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3.3.1 Constructing B

The construction of B is originally due to Momose in his paper [10], wherein he studies the images of `-adic representations of modular forms. He begins by defining a Q-vector space V that has a linear L-action and satisfies dimQL = 2. In particular, for each

prime number `, the representation ρf,` gives V` = V ⊗QQ` the structure of a GQ

-module. For each γ ∈ Γ, he constructs a Q-automorphism ηγ of V that extends to

an H-endomorphism of V`, in particular, if p is a prime number not dividing `N , then

theorem 2.16 of Momose’s paper says that

ρf,`(Frobp)ηγ= ηγρf,`(Frobp)χγ(p).

The construction of ηγ is quite complicated, and we have not developed the necessary

terminology in this thesis, so the curious reader is directed to section 2 of Momose’s paper. Finally, he gives the following theorem.

Theorem 3.3.2 (Momose, [10, 3.1]). The vector space X = P

γ∈ΓL · ηγ is a simple

central F -algebra with maximal subfield L. Moreover, it can be described as follows: for each primitive Dirichlet character of conductor r, let g(χ) =Pr−1

i=0χ(u)e2πiu/r, and

define c : Γ × Γ → L× by c(γ, σ) = g(χ−1_γ χ−1_σ )/g(χ−1_γσ). Then c is a normalised 2-cocycle in H2(Γ, L×). Then, if α ∈ L and γ, σ ∈ Γ, we have ηγησ = c(γ, σ)ηγσ and ηγα = αγηγ.

So, we have an inclusion X ,→ EndQV . Also the centraliser B of X inside EndQV is a

quaternion F -algebra.

In Ribet’s paper [15], which appeared at the same time as Momose’s paper but was written independently, he considers the case where f has weight 2, and so has an abelian variety A attached to it. In section 5, he examines the algebra X = (EndQA) ⊗ ZQ, and

determines that it is a simple central F -algebra with maximal subfield L. In particular, it is also described using the same c ∈ H2(Γ, L×) as Momose uses. Furthermore, by showing that c has order at most 2 in H2(Γ, L×) he demonstrates that X has order 1 or 2 in the Brauer group Br(L/F ) of simple central F -algebras with maximal subfield L. This means that X = M#Γ(F ) or X = M#Γ/2(D), where D is a division quaternion

F -algebra. Later, in [16], he considers the action of X upon V constructed by Momose, and demonstrates that if X = M#Γ(F ) then B = M2(F ), and otherwise B = D. So if

we know what Γ is, we have a concrete way to go about giving an explicit description of B, along with its splitting behaviour.

3.3.2 Properties of B

In this section we give some important properties of B, all of which are due to either Momose or Ribet.

Momose demonstrated that for each prime number `, the restriction of ρ` to H gives

a map

ρ`|H : H → (B ⊗QQ`)×.

From the way in which H is defined, the determinant of ρ`|H is simply χk−1_` , where χ`

is the `-adic cyclotomic character. Let H` be the image of H under ρ`. Due to Momose,

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Theorem 3.3.3 (Momose, [10, 4.1]). For all prime numbers `, the group H` is an open

subgroup of G`

Because B ⊗QQ` = M2(Q`) for almost all `, we may assume that H` is a subgroup of

A` = {x ∈ GL2(OF ⊗ZZ`) : det x ∈ Z ×(k−1)

` }

for almost all `, possibly after replacing ρ` with a conjugate.

The following theorem is due to Ribet, although Momose had already given it in the weight 2 case.

Theorem 3.3.4 (Ribet, [16, 3.1]). For all but finitely many prime numbers `, the group H` is the full group A`.

We will give an outline of Ribet’s proof.

Lemma 3.3.5 (Ribet, [13, 2.2]). Assume ` ≥ 5. Suppose H`is a subgroup of GL2(OF⊗Z

Z`) such that det : H` → Z×(k−1)_` is surjective, and the image of H` modulo ` contains

Y

λ|`

SL2(OF,λ/`OF,λ),

then H`= A`.

Lemma 3.3.6 (Ribet, [13, 3.1]). Assume ` ≥ 5. Let eH` be the image of H` under the

map Y λ|` GL2(Fλ) → Y λ|` GL2(k(λ)),

and assume that the determinant map det : eH`→ F ×(k−1)

` (recall that k is the weight of

our eigenform f ) be surjective. Further assume that eH` contains an element x such that

F`[Tr(x)2] =

Y

λ|`

k(λ),

and that the image of each projection

pλ : eHλ → GL2(k(λ))

is an irreducible subgroup with order divisible by `. Then e H`= {(uλ)λ|`∈ Y λ|` GL2(k(λ)) : det((uλ)λ|`) ∈ F ×(k−1) ` }.

It follows from these two lemmas that H` = A` whenever

i the determinant map H` → Z ×(k−1)

` surjective;

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iii there exists a finite prime p - `N of K = QH such that Tr(ρf,`(Frobp))2 generates

F ⊗QQ` as a Q`-algebra;

iv for each λ | `, the image of H` under pλ is an irreducible subgroup of GL2(k(λ))

with order divisible by `.

The `-adic cyclotomic character χ` is surjective on H for almost all `, so we can see

that (i) holds for all but finitely many `. To see a proof that (iii) holds for almost all `, see the proof of theorem 3.1 in [16]. Finally, (iv) is theorem 2.1 of [16].

3.4 Image of H

We now use B to define two algebraic groups over Q:

G(R) = {(x, λ) ∈ (B ⊗QR)×× R× : normB/Fx = λk−1},

and

G◦(R) = {(x, λ) ∈ G(R) : λ = 1}, where R denotes a Q-algebra.

Recall that if R is a topological ring then the topology on GLn(R) is the weakest

topology such that GLn(R) → Mn(R) × Mn(R); x 7→ (x, x−1) is continuous, where

Mn(R) is homeomorphic with Rn

2

in the usual way. We then give G(R) the subspace topology inherited from GLn(R) × R×

Let OB be a maximal order of B over F , let p be a prime of F , and let OB,p be the

p-adic completion of OB inside B ⊗F Fp. By replacing B with OB and Q with Z, we

may define G and G◦ as algebraic groups over Z. Our choice of OB is not unique, but we

will see now that this does not matter. If B is unramified at p, then OB,p is conjugate

to M2(OF,p)—to see this recall that OB,p fixes a lattice in Fp2—and if B is ramified at

p, then OB,pis unique—this is because there exists a unique extension of the field norm

on Fp to a multiplicative norm on B ⊗F Fp, and OB,p is then the elements with norm

≤ 1. Therefore, and our choice of O_B only alter G(bZ) up to conjugation within G(A∞_Q). Moreover, after possibly replacing ρf with a conjugate, we may assume that ρf× χ takes

values in G(bZ).

Also G is representable. To see this let us say that B has the quaternion basis basis 1, u, v, w with u2 _{= a, v}2 _{= b ∈ F . Then norm}

B/F(x) = x21− ax22− bx23+ abx24. We can

choose a and b so that they are integral and have minimal polynomials f (x), g(x) ∈ Z[x]. Finally we assume that λ is a unit, so put all together G is represented by

Z[x1, x2, x3, x4, λ, z, u, v]/(f (u), g(v), x21− ux22− vx23+ uvx24− λk−1, λz − 1).

Also recall that there exist only finitely many primes of F where B is ramified. Because the determinant of ρf|H is the (1 − k)-th power of the cyclotomic character,

the map defined by

H → GL2(OL⊗ZZ) × bb Z× Frob`7→ (ρ(Frob`), `)

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(for prime numbers ` not dividing N p) has values in G(bZ), thus ρf|H naturally gives

a map ˜ρf : H → G(bZ), and so we may state the main result of the second section of

Loeffler’s paper:

Theorem 3.4.1. The image of ˜ρf is open in G(bZ).

To prove this theorem we first need several preliminary lemmas, all of which are laid out in Loeffler’s paper [8], with essentially the same proofs as here.

Lemma 3.4.2. Let U◦ be a compact closed subgroup of G◦(bZ) such that for each prime number p the image of the projection of U◦ to G◦(Zp) is open, and for all but finitely

many prime numbers p the image of this projection is the full group G◦(Zp). Then U◦

is open in G◦(bZ).

Before we prove lemma 3.4.2 we first need a few lemmas from group theory.

Lemma 3.4.3 ([13, 2.1]). Let ` ≥ 5 be a prime number, and let K1, . . . , Kt be finite

extensions of Q`, with integer rings O1, . . . , Ot. Suppose that G is a closed subgroup of

GL2(O1) × · · · × GL2(Ot),

such that the image of G under reduction modulo ` contains S = SL2(O1/`O1) × · · · × SL2(Ot/`Ot).

Then G contains

T = SL2(O1) × · · · × SL2(Ot).

Proof. Let C be the closure of the commutator subgroup of G, then the image of C modulo ` contains the commutator subgroup of S. If the commutator group of S is simply S, then G ∩Q

iSL2(Oi) surjects onto S, and so we adapt an argument from

[17, lemma 3, IV.3.4]—wherein Serre proves that a closed subgroup X of SL2(Z`) which

surjects onto SL2(F`) must be the entire group—to show that G contains T .

So now we need only prove that S is its own commutator subgroup.and so we may assume t = 1. Let λ be a uniformiser in O1 with ramification index e, then we use

induction to show that SL2(O1/λnO1) is its own commutator subgroup for n ≤ e. If

n = 1, then we are trying to show that SL2(Fq) is its own commutator subgroup, with

q a power of `. This is shown in [1, theorem 3.4.]. So assume now that the statement is true for 1 ≤ n ≤ e − 1, and consider n + 1. We then only need to show that the commutator subgroup of SL2(O1/λn+1O1) contains the kernel of the surjection

SL2(O1/λn+1O1) → SL2(O1/λnO1).

As in the previous lemma we can identify the kernel with sl2(O1/λO1). Now given

˜

u ∈ sl2(O1/λO1), we can use the identification of sl2(O1/λO1) with the kernel to obtain

a commutator of S and u in SL2(O1/λn+1O1), which is

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which is identified with

s˜us−1− ˜u ∈ sl2(O1/λO1),

where s is the reduction of S modulo λ. Pick v ∈ sl2(O1/λO1). Then v is a linear

combination of the matrices vi satisfying v2i = 0, so we can assume v2 = 0 and moreover

that v is of the form

0 x 0 0

! .

We picked ` ≥ 5, so there exists an element a ∈ O/λO such that a26= 0, 1, so we have

v = a 0 0 a−1 ! 0 x/(a2− 1) 0 0 ! a−1 0 0 a ! − 0 x/(a 2_{− 1)} 0 0 !

as wanted. Therefore the image of v in the kernel is contained in the commutator subgroup of SL2(O1/λnO1), and so the commutator subgroup of S is S.

Definition 3.4.4. If k is a field, then PSLn(k) = SL2(k)/µn(k), where µn(k) is the

group of n-th roots of unity in k.

Remark 3.4.4.1. In algebraic geometry, one often defines PSLn to be a quotient of

algebraic groups, namely SLn/µn, however in this thesis we will only use the

group-theoretic definition given above.

Lemma 3.4.5 (Ribet). Let p ≥ 5, and let K1, . . . , Kt be finite unramified extensions

of Qp, with rings of integers O1, . . . , Ot, and residue fields k1, . . . , kt. Let G be a closed

subgroup of SL2(O1)×· · ·×SL2(Ot) which surjects onto PSL2(k1)×· · ·×PSL2(kt). Then

G = SL2(O1) × · · · × SL2(Ot).

Proof. Let H be a subgroup of SL2(k1) × · · · × SL2(kt) that surjects onto PSL2(k1) ×

· · · × PSL2(kt). Then {±1}tH = SL2(k1) × · · · × SL2(kt), so let α be a square root of

−1, then for ((−1)e1_{, . . . , (−1)}et_{) ∈ {±1}}t _{with e}

i = 0, 1 for each i let xi = α if ei = 1

and xi = 1 if ei = 0, then H contains an element of the form (±x1, . . . , ±et), which we

can square to get ((−1)e1_{, . . . , (−1)}et_{) ∈ {±1}}t_{, so H is the full group. Now because our}

extensions are unramified we have Oi/pOi = kifor each i, so we may apply lemma 3.4.3,

and the result follows.

Lemma 3.4.6. Let K be a p-adic field for some prime number p, and let Y1 and Y2 be

closed subgroups of GL2(OK) such that Y1E Y2 and Y2/Y1 is a nonabelian finite simple

group. Then Y2/Y1 is isomorphic to one of the following:

1. PSL2(Fq) where q is a power of p and q ≥ 4;

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Proof. Consider the projection map GL2(OK) → PGL2(k), where k is the residue field

of K. Let $ be a uniformiser of K. The kernel of this projection is the group

N :=    a b c d ! ∈ GL2(OK) : a − d ≡ b ≡ c ≡ 0 (mod $)    .

This group may be written as the projective limit N = lim_←−

n≥1Nn where Nn=    a b c d ! ∈ GL2(OK/$nOK) : a − d ≡ b ≡ c ≡ 0 (mod $)    .

We claim that Nn is solvable. Multiplication in Nn is given by

(x0+ x1$ + · · · + xn−1$n−1) · (y0+ y1$ + · · · + yn−1$n−1) = n−1 X i=0 i X j=0 xjyi−j$i

with x0, yi ∈ k×⊂ M2(k), xi, yj ∈ M2(k) for i, j = 1, . . . , n − 1. This allows us to write

Nn = k×× M2(k)n−1 with multiplication given as above. We then have a subnormal

series

M2(k) C M2(k)2C · · · C k×× M2(k)n−1.

Each quotient is isomorphic to either M2(k), with its usual addition, or to k×, thus Nnis

solvable. Therefore N is a projective limit of finite solvable groups, i.e. N is prosolvable. Assume Y2 is not contained in SL2(OK), then (Y2∩ SL2(OK))/(Y1 ∩ SL2(OK)) is a

normal subgroup of Y2/Y1. It is a subgroup because the map (Y2 ∩ SL2(OK))/(Y1 ∩

SL2(OK)) → Y2/Y1 given by γ(Y1∩ SL2(OK)) is an injective group homomorphism, and

the map Y2/Y1 → O_K×/ det Y1 given by γY17→ det γ det Y1has kernel Y2∩SL2(OK)/(Y1∩

SL2(OK)).

Now because N , Y1, and Y2 are closed subgroups of GL2(OK), if Y2is a subgroup of N

then Y2/Y1 is prosolvable, and because it is finite, it will also be solvable, contradicting

our assumption that Y2/Y1 is simple and nonabelian.

Now we claim that Y2∩ N is a subgroup of Y1. If not then (Y2 ∩ N )/(Y1 ∩ N ) is a

normal, proper, and non-trivial subgroup of Y2/Y1. If it were not proper then Y2/Y1

would be a subquotient of N given by closed subgroups, hence would be prosolvable. Thus Y2/Y1 ∼= (Y2/(Y1∩ N ))/(Y1/(Y1∩ N )), telling us that Y2/Y1 is a subquotient of

PSL2(k).

Let r = #k and d = gcd(2, r − 1). The result now follows from [21, §3 theorem 6.25], which lists the possible subgroups of PSL2(k), which are subgroups of the following

groups:

1. dihedral groups of order 2(r ± 1)/d;

2. groups H of order r(r − 1)/d, such that if Q is an Sylow p-subgroup of H, then Q is a normal elementary abelian subgroup—meaning that each element of Q has order p—of H such that H/Q is cyclic of order (r − 1)/d;

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3. the symmetric group on 4 elements Σ4, A4, or A5, where Andenotes the alternating

group on n elements;

4. or groups of the form PSL2(Fq) or PGL2(Fq), where r is a power of q.

The only groups on this list that are not solvable are A5, PSL2(Fq), or PGL2(Fq),

where q ≥ 4. However, by the exact sequence 1 → PSL2(Fq) → PGL2(Fq)

det

−−→ F×_q/F×2_q → 1,

we see that a nonabelian finite simple subquotient of PGL2(Fq) must be of the form A5

or PSL2(Fq0), with q0 ≥ 4, so we are done.

We are now ready to prove lemma 3.4.2.

Proof. This proof is the same as the proof given by Loeffler in [8, theorem 1.2.3], which in turn is based on the proof given by Serre in [17, §IV.3.1].

For each prime number p let U_p◦ be the image of U◦ in G◦(Zp), and let S be the set

of prime numbers p satisfying at least one of the following conditions: 1. p ≤ 5,

2. p is ramified in F/Q, 3. B is ramified at p, 4. and U_p◦6= G◦(Zp).

For a prime number p let kp be the Cartesian product Qp|pk(p), where each p is a

prime of F , and k(p) is the residue field of F at p. Then for each p /∈ S we have a natural map

U◦ → G◦(Zp) ∼= SL2(OL⊗ZZp) → PSL2(kp)

given by composition of the two natural projections. Because p /∈ S the first map is surjective, and so the map U◦ → PSL2(kp) is also surjective.

Assume for the sake of contradiction that the restriction of this map to U◦∩ G◦_(Z p)

is not surjective. Write

Q = U◦/(U◦∩ G◦(Zp)),

and define an equivalence relation ∼ on PSL2(kp) by x ∼ y if there exists u, v ∈ U◦

such that uv−1 ∈ U◦ ∩ G◦(Zp), and the images of u and v in PSL2(kp) are x and y

respectively. We find that ∼ is a congruence relation on PSL2(kp), that is to say the set

of elements x satisfying x ∼ 1 form a normal subgroup, hence PSL2(kp)/ ∼ is a group.

We now have a well-defined group homomorphism Q → PSL2(kp)/ ∼

where for each q ∈ Q we pick a representative u ∈ U◦ of q, and send u to its image in PSL2(kp). Now, if PSL2(kp)/ ∼ is the trivial group then for every x ∈ PSL2(kp)

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there exist u, v ∈ U◦ such that uv−1 ∈ G◦_(Z

p) and the image of uv−1 in PSL2(kp) is

x, contradicting our assumption that U◦∩ G◦(Zp) does not surject onto PSL2(kp). It is

well known that the groups PSL2(k(p)) are simple when p ≥ 5, see [1], and so Q must

surject onto PSL2(k(p)) for some prime number p | p of F .

However Q is the image of U◦ inQ

q6=pG ◦_(Z

q), so the finite simple group PSL2(k(p))

is a subquotient of an open compact subgroup of Q

q6=pG ◦_(Z

q).

Let K be a finite extension of F that splits B, thenQ

q6=pG◦(Zq) is a closed subgroup

of the maximal compact subgroupQ

q6=pGL2(OK⊗ZZq), and by lemma 3.4.6 this group

does not have PSL2(k(p)) as a quotient. This is because for each q the possible nonabelian

finite simple quotient groups are PSL2(F) with F a finite field of characteristic q and

order ≥ 4, or A5. Recall that the only pair of isomorphic projective special linear groups

are PSL2(F4) and PSL2(F5), and both are isomorphic to A5.

Hence U◦∩ G◦_(Z

p) surjects onto PSL2(kp), and so by lemma 3.4.5, U◦ ∩ G◦(Zp) =

G◦(Zp). This tells us that

M

p /∈S

G◦(Zp) ⊆ U◦,

and by taking the closure of this subgroup we find that Y

p /∈S

G◦(Zp) ⊆ U◦.

We now know that if the image of U◦ inQ

p∈SG◦(Zp) is open, then U◦ must be open.

Furthermore, because U◦ is a closed subgroup it suffices to show that the image of U◦ in Q

p∈SG ◦_(Z

p) has finite index. Each G◦(Zp) contains a pro-p group of finite index,

namely

Γp := Up◦∩ {x ∈ G ◦

(Zp) : x ≡ 1 (mod p)},

where U_p◦ is the image of U◦ in G◦(Zp). To see this recall that G◦(Zp) is always

isomor-phic to a subgroup of GL4(OF ⊗ZZp). Now by inducting on the size of subsets T of

S, we will show that the image of U◦ in Q

p∈T G◦(Zp) surjects onto Qp∈TΓp for every

T ⊆ S. For ease of notation we write

U_T◦ = Y p∈T Γp∩  image of U◦ in Y p∈T G◦(Zp)  .

Note that showing U_T◦ = Q

p∈T Γp is not equivalent to showing that

Q

p∈T Γp ⊆ U ◦_,

except when T = S.

We take the base case to be #T = 1, which follows immediately from how we have constructed the Γp. Assume now that T is a proper non-empty subset of S such that

U_T◦ = Q

p∈TΓp, and let T0 = T t {q} ⊆ S where q ∈ S \ T . Assume also that UT◦0 6=

Q

p∈T0Γ_p. We define an equivalence relation on Γ_q by x ∼ y if and only if there exist

u, v ∈ U_T◦∩Q

p∈T0Γpwith uv−1∈ Γq, and such that u and v map to x and y respectively.

As before, this defines a congruence relation on Γq, and so Γq/ ∼ is a group, and

Images of Adelic Representations of Modular Forms

MSc Mathematics

Master Thesis