Comptage des systèmes locaux $\ell$-adiques sur une courbe

Abstract

Let $X_1$ be a projective, smooth and geometrically connected curve over ${\mathbb{F}}_q$ with $q=p^n$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of ${\mathbb{F}}_q$. We give a formula for the number of irreducible $\ell$-adic local systems ($\ell \ne p$) with a fixed rank over $X$ fixed by the Frobenius endomorphism. We prove that this number behaves like a Lefschetz fixed point formula for a variety over ${\mathbb{F}}_q$, which generalises a result of Drinfeld in rank $2$ and proves a conjecture of Deligne. To do this, we pass to the automorphic side by Langlands correspondence, then use Arthur's non-invariant trace formula and link this number to the number of ${\mathbb{F}}_q$-points of the moduli space of stable Higgs bundles.

In French, 85 pages, final version, with minor revision from the previous version