International audienceLet 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 étale 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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International audienc

We compute the equations of the Frobenius action on semi-stable rank 2 vector bundles with trivial determinant over a supersingular, proper and smooth, curve of genus 2, defined over an algebraically closed field of characteristic 2 and draw some geometric consequences. The computation strategy is to deform the situation to an ordinary curve where we can use the results of Y. Laszlo and C. Pauly.