Serre’s conjecture. We fix an algebraic closure Q and an embedding Q ‹→ C. We write c ∈ GQ for the element corresponding to complex conjugation. Suppose ρ : GQ → GL2(Fp) is any mod p Galois representation, then we say ρ is odd if det(ρ(c)) = −1. We first recall the weak form of Serre’s conjecture as stated in [Ser73]. Theorem 1.1 (The weak form of Serre’s conjecture). Given a continuous, irreducible and odd representation ρ : GQ → GL2(Fp), then there exists a cuspidal eigenform f such that ρ ∼= ρf . If this is the case and f has weight k and level N , we say ρ is modular of weight k and level N . In [Ser87], Serre refined this statement to include explicit data of the modular form such that this holds, now known as the strong form of Serre’s conjecture: Theorem 1.2 (The strong form of Serre’s conjecture). Suppose ρ : GQ → GL2(Fp) is continuous, irreducible and odd. Then there exists a cuspidal eigenform of weight k(ρ), level N (ρ) and character ϵ(ρ) such that ρ ∼= ρf . Moreover, k(ρ) and N (ρ) are the minimal weight and level such that this holds. The value N (ρ) is the prime-to-p Artin conductor of ρ. In Chapter 2 we care- fully study the recipe for the weight k(ρ). We will not focus on the character ϵ(ρ). The minimality statement for the weight and level means the following: if ρ =∼ ρf' for some f′ of weight k′ ≥ 2 and N′ prime to p, then N′ is a multiple of N and k′ ≥ x. Xxxxx’x conjecture was proven by Xxxxx, Xxxxxxxxxxxx and Xxxxx in [KW09a], [KW09b] and [Kis09b] building on the work of many others. The proof relied on preceding work proving equivalence between the weak form and the strong form of the conjecture. In particular, it involved a result by Xxxxxxxxx, which is known as the weight part of Serre’s conjecture ( [Edi92, Theorem 4.5]): Theorem 1.3 (The weight part of Serre’s conjecture). Let ρ : GQ → GL2(Fp) be continuous, irreducible and odd. Suppose we have a cuspidal eigenform g of type (N, k, ϵ) with p - N such that ρ ∼= ρg. Then
Serre’s conjecture. Let us start with the fundamental theorem from Xxxxxxx’s paper [Del71]. Let k ě 2 and N ě 1 be integers and suppose f < Sk(Γ1(N ), χ), that is, we let f be a cuspform for Γ1(N ) with character χ. Write the q-expansion of f as f = ně1 anq . We suppose, moreover, that f is an eigenvector for the Hecke operators which is normalised such that a1 = 1. The coefficients an are algebraic integers. If we denote the ring of algebraic integers by Z, we can pick an ideal p above p. We identify the reduction Z/p with Fp and we may therefore consider reductions of algebraic integers to lie in Fp. Then by the process described in the previous section, we immediately get the following theorem.
