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

TL;DR

This paper investigates the fixed points of the Verschiebung operator on certain moduli spaces of genus-2 curves in characteristic 2, revealing new geometric insights into their structure and monodromy representations.

Contribution

It provides a detailed analysis of the fixed point locus of the Verschiebung on M_x(2,0) for genus-2 curves in characteristic 2, connecting it to monodromy representations.

Findings

Existence of infinite image representations of π_1(X) in SL(2,κ[s])

Geometric interpretation of Laszlo's counterexample on monodromy finiteness

Characterization of automorphism group for the curves studied

Abstract

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].