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 $Pic^0$.

中文翻译：

---

几何阿贝尔基本群的 p-adic 曲线和截面的算术

令 $X$ 是 $g(X)\ge 1$ 属在 $p$-adic 局部域上的适当、平滑且几何连接的曲线。我们证明存在一个有效可计算的开放仿射子方案 $U\subset X$，其性质为 $period (X)=1$，并且 $index (X)$ 等于 $1$ 或 $2$（分别为 $period(X) )=index (X)=1$，假设 $period (X)=index (X)$)，如果（分别当且仅当）$U$ 的几何阿贝尔基本群的精确序列分裂。我们计算与 $X$ 相关的几何阿贝尔绝对伽罗瓦群的精确序列的分裂的torsor，并给出了与 $Pic 正交的 $p$-adic 局部场上曲线的算术基本群的截面的新特征^0$（分别为 $Pic^{\wedge}$）。