Radius of convergence of p-adic connections: an application to the p-adic Rolle theorem

We illustrate the theory of the radius of convergence of a connection on a p-adic curve X, by deducing from it a simple proof of a variant of Alain Robert's p-adic Rolle theorem. We need to carefully compare our global notion of radius of convergence, depending on the choice of a semistable formal model of X, and the local intrinsic notion of radius of convergence at a point x of Berkovich type 2 or 3, of Kedlaya. (Both notions go back to Dwork, Robba, Christol,...). The coincidence of the two notions when x is a point of the skeleton of the chosen semistable formal model of X, is crucial in the conclusion of our proof. The same method applies to the discussion of the p-adic geometric ramification locus, in the sense of Berkovich, of an etale covering of smooth p-adic curves.