Iwasawa theory began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to Selmer groups of elliptic curves (Abelian varieties in general). At the turn of this century, Coates and Sujatha initiated the study of a subgroup of the Selmer group of an elliptic curve called the \textit{fine} Selmer group.
The focus of this thesis is to understand arithmetic properties of this subgroup. In particular, we understand the structure of fine Selmer groups and their growth patterns. We investigate a strong analogy between the growth of the p-rank of the fine Selmer group and the growth of the p-rank of the class groups. This is done in the classical Iwasawa theoretic setting of (multiple) \ZZ_p-extensions; but what is more striking is that this analogy can be extended to non-p-adic analytic extensions as well, where standard Iwasawa theoretic tools fail.
Coates and Sujatha proposed two conjectures on the structure of the fine Selmer groups. Conjecture A is viewed as a generalization of the classical Iwasawa μ=0 conjecture to the context of the motive associated to an elliptic curve; whereas Conjecture B is in the spirit of generalising Greenberg's pseudonullity conjecture to elliptic curves. We provide new evidence towards these two conjectures.Ph.D
