论文标题
$ p $ adadic曲线的算术和几何曲线的部分
Arithmetic of $p$-adic curves and sections of geometrically abelian fundamental groups
论文作者
论文摘要
令$ x $为$ g（x）\ ge 1 $在$ p $ adiC的本地领域上的合适，平滑且几何连接的曲线。我们证明，存在一个有效的可计算开放式仿射$ u \ u \子集x $，其属性属性（x）= 1 $，$ index（x）$等于$ 1 $或$ 2 $或$ 2 $（resp。$ ervise（x）= index（x）= index（x）= 1 $，$ necy（x）= index（x）= index（x）$ fult。 $ u $拆分。我们计算了与$ x $相关的几何Abelian Ablesute Galois组的精确顺序的分裂的Torsor，并给出了在$ p $ p $ - adic的本地领域的算术基本组的新表征，这些曲线是$ p $ - adic的本地领域，这些曲线是$ pic^0 $ pic^0 $ pic^$ pic^$ pic^^{\ wedge \ wedge} $）。结果，我们观察到由Hoshi在[Hoshi]中构建的非几何（几何pro-P $）部分是正交的$ pic^0 $。
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 $Pic^0$.
