The Fixed Point Locus of the Verschiebung on M_x(2,0) for Genus-2 Curves X in Charateristic 2

In this note, we prove that for every ordinary genus-2 curve $X$ over a finite field $\kappa$ of characteristic 2 with $\text{Aut}(X\mathrm{/}\kappa )=\text{db}Z\mathrm{/}2\text{db}Z\times S_3$, there exist $\text{SL}(2,\kappa \text{sembrack}s)$-representations of ${\pi }_1(X)$ such that the image of ${\pi }_1(\bar{X})$ is infinite. This result gives a geometric interpretation of Laszlo’s counterexample [12] to a question regarding the finiteness of the geometric monodromy of representations of the fundamental group [4].