On De Jong’s conjecture

Abstract

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q.

Assume that $\rho \left| {_{\pi (\bar X)} } \right.$ 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 ρ 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) ≠ 2, thereby proving de Jong’s conjecture in this case.