Pro-$p$ Criterion of Good Reduction of Punctured Elliptic Curves

Abstract

In this paper we consider an affine hyperbolic curve $X$ over a $p$-adic local field $K$ obtained from an elliptic curve $E$ by removing a $K$-rational point given by the origin $O$. We introduce the maximal geometrically pro-$p$ quotient ${\Pi }_X$ of the étale fundamental group ${\pi }_1^{\acute{\mathrm{e}}\mathrm{t}}(X)$ of the hyperbolic curve $X$ and analyse the problem of determining the reduction type of the elliptic curve $E$ from the topological group ${\Pi }_X$. As our main result we give a group-theoretic construction of the reduction type of $E$ from the group ${\Pi }_X$ under the assumption that $p\ge 5$ and that $E$ has a nontrivial $K$-rational $p$-torsion point.