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 $K_{\frak p}$-valued points of the modular Jacobians for a ${\frak p}$-adic completion $K_{\frak p}$ for a prime $\frak p|p$ of $K$. As a consequence, if $K_{\frak p}=\Bbb Q_p$, we give an exact formula for the rank of the ordinary/co-ordinary part of the closure.