Local sections of arithmetic fundamental groups of p-adic curves

2023/10/30 by Saidi, Mohamed
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2310.19556

Abstract

We investigate sections of the arithmetic fundamental group pi₁(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 pi₁(X) splits, then index (Y)=1. We also exhibit a necessary and sufficient condition for a section of pi₁(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.

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