Séminaire de Théorie des Nombres de Montpellier :

Le 06 décembre 2004 à None -

Présentée par Osserman Brian - MSRI, Berkeley

« Frobenius-unstable vector bundles and the generalized Verschiebung »

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 discuss the generalized Verschiebung map $V_r:M_r(C^{(p)}) \dashrightarrow M_r(C)$ induced by pulling back under Frobenius. We begin with a survey of general background results on the Verschiebung, and state a theorem on its geometry for $r=2, g=2$. We then discuss the relationship between Frobenius-unstable bundles and the theory of connections with vanishing $p$-curvature, with a focus on the case $r=2$ and Mochizuki's work in that context.