On mod 3 Galois representations with conductor 4

Abstract

Let GQ be the absolute Galois group Gal(Q=Q) of Q. Let Fp be an algebraic closure of the flnite fleldFp of p elements. In this paper, we prove the non-existence of certain mod 3 Galois representation: Theorem 1. There exist no irreducible representations ‰ : GQ ! GL2(F3) with N(‰) dividing 4. Here, N(‰) = Q p-3 p n p(‰) is the Artin conductor of ‰ outside 3 ([6], x1.2; the deflnition of the exponent np(‰) will be recalled below). This proves a special case of Serre’s conjecture ([6]). Indeed, the conjecture predicts that such a representation, up to twist by a power of the mod 3 cyclotomic character, come from a cuspidal eigenform of level 4 and weight • 4, but there are no such forms. Such a result may serve as the flrst step of an inductive proof of Serre’s conjecture for N(‰) = 4 if Khare’s proof in the case of N(‰) = 1 ([3]) can be extended. Serre’s conjecture is known to be true if the image Im(‰) of ‰ is solvable ([4], Thm. 4). So, it remains for us to prove the Theorem 1 in the following two cases: (i) Im(‰) is non-solvable, (ii) ‰ is even and Im(‰) is solvable.