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

Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ 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. $Pic^{\wedge}$). As a consequence we observe that the non-geometric (geometrically pro-$p$) section constructed by Hoshi in [Hoshi] is orthogonal to $Pic^0$.