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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 aneffectively computable open affine subscheme $U\subset X$ with the propertythat $period (X)=1$, 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 exactsequence of the geometrically abelian fundamental group of $U$ splits. Wecompute the torsor of splittings of the exact sequence of the geometricallyabelian absolute Galois group associated to $X$, and give a newcharacterisation 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 $Pic^0$.

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