The conjecture

Let us now state the conjecture. It has a weak form and a strong form.

Conjecture 1.1(Serre’s conjecture, weak form, [70, Conjecture 3.2.3]). Consider an odd irreducible representation ρ : Gal(Q/Q)→ GL2(F`). Then there exists a newform f of

some level and some weight, a primeλ of Kf above`and an embeddingFλ ,→F`such that

ρ∼=ρ_f,λ holds, where we viewρ_f,λ as a representation over_{F`</sub>via the embedding<sub>Fλ</sub> ,→<sub>F`}.

Conjecture 1.2(Serre’s conjecture, strong form, [70, Conjecture 3.2.6]). In the notation and statement of Conjecture 1.1 there exists an f of level dividing N(ρ)and weight k(ρ).

It is a result of many people that the weak version is equivalent to the strong version; instead of compiling a complete list of names here, we refer to the overview article [42]. Serre’s conjecture has been proven subsequently for level one in [38], for representations of odd level over fields of odd characteristic in [39] and finally in general in [43]. In all cases, the main ideas originate from the proof of the modularity theorem for elliptic curves by Taylor and Wiles [86].

Chapter 2

Computations with modular forms

In this chapter we will discuss several aspects of computations with modular forms. Let us warn the reader on beforehand that we will focus on how to compute in practice, not on theoretical aspects of computability. What in theory can be proven to be computable, can often not be computed in practice and what in practice can be computed, can often not be proven to be computable in theory.

2.1

Modular symbols

Modular symbols provide a way of doing symbolic calculations with modular forms, as well as the homology of modular curves. In this section as well, our intention is to give the reader an idea of what is going on rather than a complete and detailed account of the material. For more details and further reading on the subject of modular symbols, the reader could take a look at [51], [72] and [53]. A computational approach to the material can be found in [78] and [79].

2.1.1

Definitions

Let A be the free abelian group on the symbols {α,β} with α,β ∈_P1(_Q). Consider the

subgroupI⊂Agenerated by all elements of the forms

{α,β}+{β,γ}+{γ,α}, {α,β}+{β,α}, and {α,α}.

We define the group

M2:= (A/I)/torsion

as the quotient ofA/I by its torsion subgroup. By a slight abuse of notation, we will denote the class of{α,β}in this quotient also by{α,β}. We have an action GL+₂(_Q)on_M2by

γ{α,β}:={γ α,γ β},

whereγ acts on_P1(_Q)by fractional linear transformations.

Fork≥2, we consider also the abelian group _Z[x,y]_k−2⊂_Z[x,y] of homogeneous polyno- mials of degreek−2 and we let matrices in GL+₂(_Q) with integer coefficients act on it on the left by a c b d P(x,y):=P(dx−by,−cx+ay). We define Mk:=Z[x,y]k−2⊗M2,

and we equip Mk with the component-wise action of integral matrices in GL+2(Q) (that is

γ(P⊗α) =γ(P)⊗γ(α)).

Definition 2.1. Letk≥2 be an integer. LetΓ⊂SL2(Z)be a subgroup of finite index and let

I⊂_Mkbe the subgroup generated by all elements of the formγx−xwithγ∈Γandx∈_Mk.

Then we define the space ofmodular symbolsof weightk forΓto be the quotient ofMk/I

by its torsion subgroup and we denote this space byMk(Γ):

Mk(Γ):= (Mk/I)/torsion.

In the special caseΓ=Γ1(N), which we will mostly be interested in, Mk(Γ)is called the

space of modular symbols of weightk and level N. The class of{α,β} inMk(Γ) will be

denoted by{α,β}Γor, if no confusion exists, by{α,β}.

The group Γ0(N) acts naturally on Mk(Γ1(N)) and hence induces an action of (Z/NZ)×

on _Mk(Γ1(N)). We denote this action by the diamond symbol hdi. The operator hdi on

Mk(Γ1(N))is called adiamond operator. This leads to the notion of modular symbols with character.

Definition 2.2. Let ε :(_Z/N_Z)× →_C× be a Dirichlet character. Denote by_Z[ε]⊂_Cthe

subring generated by all values of ε. Let I ⊂_Mk(Γ1(N))⊗Z[ε] be the _Z[ε]-submodule

generated by all elements of the formhdix−ε(d)xwithd∈(Z/NZ)× andx∈Mk(Γ1(N)). Then we define the space_Mk(N,ε)of modular symbols of weightk, levelN and characterε

as the_Z[ε]-module

Mk(N,ε):= Mk(Γ1(N))⊗Z[ε]/I/torsion.

We denote the elements of_Mk(N,ε)by {α,β}N,ε or simply by{α,β}. Ifε is trivial, then

we haveMk(N,ε)=∼Mk(Γ0(N)).

Let_B2be the free abelian group on the symbols{α}withα ∈P1(Q)with action of SL2(Z)

byγ{α}={γ α}and define_Bk:=Z[x,y]k−2⊗B2with component-wise SL2(Z)-action. El-

ements of_Bk are calledboundary modular symbols. For a subgroup Γ<SL2(Z) of finite

index, we define_Bk(Γ)as

Bk(Γ):= (Bk/I)/torsion

where I is the subgroup of _Bk generated by all elements γx−x with γ ∈ Γ and x ∈Bk.

where I is the_Z[ε]-submodule of Bk(Γ1(N))⊗Z[ε] generated by the elements γx−ε(γ)x

withγ ∈Γ0(N).

We haveboundary homomorphismsδ :_Mk(Γ)→Bk(Γ)andδ :Mk(N,ε)→Bk(N,ε), de-

fined by

δ(P⊗ {α,β}) =P⊗ {β} −P⊗ {α}.

The spaces of cuspidal modular symbols, denoted by _Sk(Γ) and Sk(N,ε) respectively are

defined as the kernel ofδ.