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

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

2011-08-08

arxiv.org/abs/1108.1633v1

Mathematics

Number Theory

This note originated from reading Xander Fabers's two papers "Topology and geometry of the Berkovich ramification locus for ra

Scientific paper

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.

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