Local sections of arithmetic fundamental groups of p-adic curves

TL;DR

This paper studies sections of the arithmetic fundamental group of p-adic curves, showing they can be lifted to cuspidally abelian Galois groups and providing conditions for sections to come from rational points.

Contribution

It proves the liftability of sections to cuspidally abelian Galois groups and characterizes when sections originate from rational points on the compactification.

Findings

Sections can be unconditionally lifted to cuspidally abelian Galois groups.

If the fundamental group sequence splits, the index of the compactification is 1.

The Picard group of an affinoid p-adic curve is finite.

Abstract

We investigate sections of the arithmetic fundamental group pi_1(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 pi_1(X) splits, then index (Y)=1. We also exhibit a necessary and sufficient condition for a section of pi_1(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.