A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)

TL;DR

This paper discusses Deligne's finiteness theorem for Galois representations over function fields, establishing the finiteness of certain sheaves and the existence of a number field containing all Frobenius traces, based on Lafforgue's Langlands correspondence.

Contribution

It provides a detailed account of Deligne's proof of finiteness of irreducible lisse sheaves with bounded ramification over finite fields, leveraging Lafforgue's Langlands correspondence.

Findings

Finiteness of irreducible lisse sheaves with bounded ramification

Existence of affine moduli of finite type over Q

Number field containing all Frobenius traces

Abstract

Revised: just some typos, reorganized a bit the article. It will be published in the VIASM Annual meeting, Hanoi. We give a detailed account of Deligne's letter to Drinfeld dated June 18, 2011, in which he shows that there are finitely many irreducible lisse $\bar \Q_\ell$-sheaves with bounded ramification, up to isomorphism and up to twist, on a smooth variety defined over a finite field. The proof relies on Lafforgue's Langlands correspondence over curves. In addition, Deligne shows the existence of affine moduli of finite type over $\mathbb{Q}$. A corollary of Deligne's finiteness theorem is the existence of a number field which contains all traces of the Frobenii at closed points, which was the main result of his recent article and which answers positively his own conjecture from Weil II.