In this note, we prove that for every ordinary genus-2 curve $X$ over a finite field $\kappa$ of characteristic 2 with $\text{ Aut}(X/\kappa)=\db{ Z}/2\db{ Z} \times S_3$, there exist $\text{ SL}(2,\kappa\sembrack{ s})$-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].
Categories: math.AG