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.
Categories: math.AG math.NT