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

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 $\bar{X}$ and $\bar{S}$ be their base changes to an algebraic closure of ${\mathbb{F}}_q$. We study the number of $\mathrm{\ell }$-adic local systems $(\mathrm{\ell }\mathrm{\ne }p)$ in rank $2$ over $\bar{X}-\bar{S}$ with all possible prescribed tame local monodromies fixed by $k$-fold iterated action of Frobenius endomorphism for every $k\ge 1$. 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.