Rank 2 $\ell$-adic local systems and Higgs bundles over a curve

TL;DR

This paper counts rank 2 $ ext{ell}$-adic local systems with prescribed monodromies over a curve over a finite field, confirming Deligne's conjectures and linking the counts to Higgs bundle moduli.

Contribution

It provides a counting formula for rank 2 $ ext{ell}$-adic local systems with fixed monodromies, confirming Deligne's conjectures and relating the counts to Higgs bundles.

Findings

Confirmed Deligne's conjectures on local system counts.

Expressed counts in terms of Higgs bundle moduli.

Validated the Lefschetz fixed point formula analogy.

Abstract

Let $X$ be a smooth, projective, and geometrically connected curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline{X}$ and $\overline{S}$ be their base changes to an algebraic closure of $\mathbb{F}_q$. We study the number of $\ell$-adic local systems $(\ell\neq p)$ in rank $2$ over $\overline{X}-\overline{S}$ with all possible prescribed tame local monodromies fixed by $k$-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.