Abstract
AbstractWe investigatesectionsof the arithmetic fundamental group$\pi _1(X)$whereXis either asmooth affinoid p-adic curve, or aformal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, ifXadmits a compactificationY, and the exact sequence of$\pi _1(X)$ splits, then$\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of$\pi _1(X)$to arise from arational pointofY. One of the key ingredients in our investigation is the fact, we prove in this paper in caseXis affinoid, that the Picard group ofXisfinite.
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