The generalized Verschiebung map for curves of genus 2

Abstract

Let C be a smooth curve, and M r (C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map $$V_{r}: M_{r}(C^{(p)}) \dashrightarrow M_{r}(C)$$ induced by pulling back under Frobenius. Our main result is a computation of the degree of V 2 for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.