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# The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus-2 Curves $X$ in Charateristic $2$

Published:2013-08-10
Printed: Jun 2014
• YanHong Yang,
Department of Mathematics, Columbia University, New York, NY 10027, USA
 Format: LaTeX MathJax PDF

## Abstract

We prove that for every ordinary genus-$2$ curve $X$ over a finite field $\kappa$ of characteristic $2$ with $\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist $\textrm{SL}(2,\kappa[\![s]\!])$-representations of $\pi_1(X)$ such that the image of $\pi_1(\overline{X})$ is infinite. This result produces a family of examples similar to Laszlo's counterexample to de Jong's question regarding the finiteness of the geometric monodromy of representations of the fundamental group.
 Keywords: vector bundle, Frobenius pullback, representation, etale fundamental group
 MSC Classifications: 14H60 - Vector bundles on curves and their moduli [See also 14D20, 14F05] 14D05 - Structure of families (Picard-Lefschetz, monodromy, etc.) 14G15 - Finite ground fields

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