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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
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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 vector bundle, Frobenius pullback, representation, etale fundamental group
MSC Classifications: 14H60, 14D05, 14G15 show english descriptions Vector bundles on curves and their moduli [See also 14D20, 14F05]
Structure of families (Picard-Lefschetz, monodromy, etc.)
Finite ground fields
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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