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 $\stackrel{̲}{\rho }:G_Q\to {\mathrm{GL}}_2({\stackrel{̲}{F}}_{\ell })$ which is odd arises from a newform. Here $G_Q$ is the absolute Galois group of $Q$, and ${\stackrel{̲}{F}}_{\ell }$ is an algebraic closure of the finite field $F_{\ell }$ of $\ell$ of ℓ elements. We formulate such a conjecture for $n$-dimensional mod ℓ representations of ${\pi }_1(X)$ for any positive integer $n$ and where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($p\ne \ell$), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for $\ell >2$ follows from a result announced by Gaitsgory in [G]. The methods are different.