$\ell$-adic local systems and Higgs bundles: the generic case

Abstract

Let $X$ be a projective smooth geometrically connected curve defined over a finite field $\mathbb{F}_q$ of cardinality $q$. Let $S$ be a finite set of closed points of $X$. Let $\bar{X}$ and $\bar{S}$ be the base change of $X$, $S$ to an algebraic closure. We consider the set of $\ell$-adic ($\ell\nmid q$) local systems of rank $n$ over $\bar{X}-\bar{S}$ with prescribed tame regular semisimple and generic ramifications in $\bar{S}$. The genericity ensures that such an $\ell$-adic local system is automatically irreducible. We show that the number of these $\ell$-adic local systems fixed by Frobenius endomorphism equals the number of stable logarithmic Higgs bundles of rank $n$ and degree $e$ coprime to $n$, with a fixed residue, up to a power of $q$. In the split case, this number is equal to the number of stable parabolic Higgs bundles (with full flag structures) fixed by $\mathbb{G}_m$-action with generic parabolic weights.