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

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

2008-03-10

arxiv.org/abs/0803.1380v2

Mathematics

Algebraic Geometry

23 pages, revised version, to appear in Ann. Inst. Fourier vol. 59 (2009)

Scientific paper

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.

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