On de Jong's conjecture

TL;DR

This paper sketches a proof of de Jong's conjecture for smooth projective curves over finite fields, linking it to the geometric Langlands conjecture and automorphic forms, under the assumption that the characteristic is not 2.

Contribution

It provides a proof of de Jong's conjecture by establishing a version of the geometric Langlands conjecture assuming char(F) ≠ 2.

Findings

De Jong's conjecture holds when char(F) ≠ 2.

The proof connects automorphic forms with Galois representations.

A version of the geometric Langlands conjecture is validated in this context.

Abstract

Let $X$ be a smooth projective curve over a finite field $F_q$. Let $\rho$ be a continuous representation $\pi(X)\to GL_n(F)$, 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 $char(F)\neq 2$, thereby proving de Jong's conjecture in this case.