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

Hélène Esnault, Moritz Kerz

Published 2012-08-01, updated 2012-11-29Version 3

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.