#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 221-264

** Haruzo Hida **
Limit Mordell--Weil Groups and their p-Adic Closure

This is a twin article of \cite{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 $K_\pG$-valued points of
the modular Jacobians for a $\pG$-adic completion $K_\pG$ for a prime
$\pG|p$ of $K$. As a consequence, if $K_\pG=\Qp$, we give an exact
formula for the rank of the ordinary/co-ordinary part of the closure.

2010 Mathematics Subject Classification: primary: 11F25, 11F32, 11G18, 14H40; secondary: 11D45, 11G05, 11G10

Keywords and Phrases: modular curve, Hecke algebra, modular deformation, analytic family of modular
forms, Mordell--Weil group, modular jacobian

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