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Let $X$ be a smooth, projective, and geometrically connected curve definedover a finite field $\mathbb{F}_q$ and $S\subseteq X$ a subset of closedpoints. Let $\bar{X}$ and $\bar{S}$ be their base changes to an algebraicclosure of $\mathbb{F}_q$. We study the number of $\ell$-adic local systems inrank $2$ over $\bar{X}-\bar{S}$ with prescribed tame local monodromies fixed by$k$-fold iterated action of Frobenius endomorphism for every $k\geq 1$. Weconfirm some conjectures of Deligne predicting that these numbers behave as ifthey were obtained from a Lefschetz fixed point formula. In fact, in all cases,our counting results are expressed in terms of the numbers of some Higgsbundles.

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