Structure of ordinary $\Lambda$-adic arithmetic cohomology groups

Abstract

We study the $\Lambda$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $\Lambda$-adic Selmer groups coming from two different towers have "almost same" $\Lambda$-module structures. We also prove the cotorsionness of $\Lambda$-adic Tate-Shafarevich group under mild assumptions.