Radius of Convergence of p-adic Connections and the p-adic Rolle Theorem

We apply the theory of the radius of convergence of a p-adic connection [2] to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. We detail in sections 1 and 2, how to obtain convergence estimates for the radii of convergence of analytic sections of such a finite morphism. In the case of an étale covering of curves with good reduction, we get a lower bound for that radius, corollary 3.3, and obtain, via corollary 3.7, a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich, theorem 0.2. We take this opportunity to clarify the relation between the notion of radius of convergence used in [2] and the more intrinsic one used by Kedlaya [16, Def. 9.4.7.]. © 2013 Springer Basel.