On de Jong's conjecture

Abstract

Let $X$ be a smooth projective curve over a finite field $F_q$. Let $\rho$ be a continuous representation $\pi (X)\to G{L}_n(F)$, where $F=F_l((t))$ with $F_l$ being another finite field of order prime to $q$. Assume that $\rho {|}_{\pi (\bar{X})}$ is irreducible. De Jong's conjecture says that in this case $\rho (\pi (\bar{X}))$ is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an $F$-valued automorphic form corresponding to $\rho$ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that $char(F)\ne 2$, thereby proving de Jong's conjecture in this case.