Local sections of arithmetic fundamental groups of p-adic curves

Abstract

We investigate sections of the arithmetic fundamental group π1(X) where X is either a smooth affinoid p-adic curve, or a formal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of π1(X) splits, then index(Y ) = 1. We also exhibit a necessary and sufficient condition for a section of π1(X) to arise from a rational point of Y . One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite.