Limit Mordell-Weil groups and their $p$-adic closure

Abstract

This is a twin article of [H14b], where we study the projective limit of the Mordell-Weil groups (called pro $\Lambda$-MW groups) of modular Jacobians of $p$-power level. We prove a control theorem of an ind-version of the $K$-rational $\Lambda$-MW group for a number field $K$. In addition, we study its $p$-adic closure in the group of $K_{\mathfrak{p}}$-valued points of the modular Jacobians for a $\mathfrak{p}$-adic completion $K_{\mathfrak{p}}$ for a prime $\mathfrak{p}|p$ of $K$. As a consequence, if $K_{\mathfrak{p}}={\mathbb{Q}}_p$, we give an exact formula for the rank of the ordinary/co-ordinary part of the closure.