Past lectures

Abstract

Let k and N be integers ≥ 1. Let Γ1(N) ⊂ SL2(Z) be the congruence subgroup of matrices 􏰀a b􏰁 ∈ SL2(Z),c ≡ 0mod.N,a ≡ d ≡ 1mod.N. Let f be a primitive modular form of weight

cd

k for Γ1(N). Let an(f) be the Fourier coefficients of f. The field Ef generated by the an(f) is a finite extension of the rationals Q. For each prime p and each embedding ip of Ef in Qp, Deligne and Serre, using etale cohomology, have constructed a representation ρip(f) of the Galois group GQ of Q in GL2(Qp), which is unramified outside N and p. It is characterized by the Eichler- Shimura relation : tr(ρip(Frobl)) = ip(al(f)), for l prime to pN. Given f, the collection of the ρip(f) forms a compatible system of representations of GQ. The ρip(f) are odd, meaning that the image of the complex conjugation c has eigenvalues 1 and −1. We call ρ ̄ip(f) the reduction modulo p of ρip(f). It is well defined up to semisimplification ; it has finite image.

Serre’s conjecture states that every odd absolutely irreducible representation GQ → SL2(F), F finite field of characteristic p, is modular i.e. arises as above as the reduction of the Galois representation attached to a primitive modular form.

The conjecture has been recently proved by Khare and the lecturer, building on previous works of many mathematicians.

The purpose of the course is to give an outline of the signification of the conjecture, and a related conjecture of Fontaine and Mazur which characterizes the p-adic representations ρ which are isomorphic to a ρip(f). The representation ρ should be “geometric”, a condition on the ramification of ρ.

We intend to present the main tools which enter in the proof. More precisely, we will present the construction of ρip(f) when f is of weight 2. It follows from congruences between modular forms that any modular ρ ̄ arises from an f of weight 2. Given ρ ̄, Serre defines a weight k(ρ ̄) and a level N(ρ ̄) such that there should exist an f of weight k(ρ ̄) and level N(ρ ̄) giving rise to ρ ̄. This stronger form of the conjecture has concrete consequences. It is a result of the work of many mathematicians that this stronger form of the conjecture is a consequence of the existence conjecture. We will give the strategy of the proof of the existence conjecture, which is a recurrence on p,k(ρ ̄) and N(ρ ̄). It uses lifting modularity theorems of the kind : ρ ̄ modular and ρ geometric implies that ρ is modular. These theorems are extensions of a theorem of Wiles, which allowed to prove that every elliptic curve defined over Q is modular, and Fermat theorem. Another tool is the theorem of Taylor giving the potential modularity of a non necessarily modular ρ ̄. It allows to prove the existence of a p-adic lifting ρ of a modulo p Galois representation ρ ̄ with control on the ramification. Furthermore, it allows to prove that ρ is a member of a compatible system.