Mod ℓ representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields

Abstract

There is a well-known conjecture of Serre that any continuous, irreducible representation ρ̲:GQ→GL2(F̲ℓ) which is odd arises from a newform. Here GQ is the absolute Galois group of Q, and F̲ℓ is an algebraic closure of the finite field Fℓ of ℓ of ℓ elements. We formulate such a conjecture for n-dimensional mod ℓ representations of π1(X) for any positive integer n and where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p (p≠ℓ), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for ℓ>2 follows from a result announced by Gaitsgory in [G]. The methods are different.