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Comptage des systèmes locaux $\ell$-adiques sur une courbe (1807.04659v5)

Published 12 Jul 2018 in math.AG and math.NT

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 ($\ell\neq p$) 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.

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