UCLA

In this thesis, we study the structure of various arithmetic cohomology groups as Iwasawa

modules, made out of two different $p$-adic variations. In the first part, for an abelian variety

over a number field and a $\mathbb{Z}_p$-extension, we study the relation between structure of

the Mordell-Weil, Selmer and Tate-Shafarevich groups over the $\mathbb{Z}_p$-extension as Iwasawa

modules.

In the second part, we consider the tower of modular curves, which is an analogue of the

$\mathbb{Z}_p$-extension in the first part. We study the structure of the ordinary parts of the arithmetic

cohomology groups of modular Jacobians made out of this tower. We prove that the ordinary

parts of $\Lambda$-adic Selmer groups coming from one chosen tower and its dual tower have almost

the same $\Lambda$-module structures. This relation of the two Iwasawa modules explains well the

functional equation of the corresponding $p$-adic $L$-function. We also prove the cotorsionness

of $\Lambda$-adic Tate-Shafarevich group under mild assumptions.