Rank 2 $\ell$-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 $\overline{X}$ 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.