Mod $\ell$ representations of arithmetic fundamental groups: Part I (An analog of Serre's conjecture for function fields)

Abstract

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the original conjecture of Serre is wide open at the moment, in this paper we prove the function field analog of Serre's conjecture only assuming largeness of image of the representation and some assumptions on the ramification of the representation (for example if the mod l representation is unramified and has full image, l is odd and the dimension and characteristic of the representation are coprime).