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

Let X be a proper, smooth, and geometrically connected curve of genus 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 , and index(X) equals 1 or 2 (resp. period(X)=index(X)=1 , assuming period(X)=index(X) ), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of U splits . 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 .