The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic

TL;DR

This paper explicitly computes the Frobenius action on rank 2 semi-stable vector bundles over genus 2 and 3 curves in small characteristic, revealing deep links with elliptic and abelian varieties.

Contribution

It provides explicit descriptions of Frobenius action on vector bundles for small genus curves in characteristic 3 to 7, connecting it to Prym varieties and invariant lines or Kummer surfaces.

Findings

Explicit equations for Frobenius action in genus 2 and 3 cases.

Identification of invariant lines and Kummer surfaces related to Prym varieties.

Computational methods for Frobenius action in small characteristic.

Abstract

Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double \'etale coverings of X. We are therefore able to compute explicit equations of this action in these cases. We perform some of these computations and draw some consequences.