Arithmetic of p-adic curves and sections of geometrically abelian fundamental groups

Abstract

Let X be a proper, smooth, and geometrically connected curve of genus g(X)≥1 g(X)≥1 over a p-adic local field. We prove that there exists an effectively computable open affine subscheme U⊂X with the property that period(X)=1 period(X)=1, and index(X) index(X) equals 1 or 2 (resp. period(X)=index(X)=1 period(X)=index(X)=1, assuming period(X)=index(X) period(X)=index(X), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of Usplits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to X, and give a new characterisation of sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to Pic 0 (resp. Pic ∧). As a consequence we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi [3] is orthogonal to Pic 0.