On the Cyclotomic Iwasawa Invariants of Elliptic Curves

Abstract

Iwasawa theory began with investigations into the Galois module structure of ideal class groups, originally developed by Kenkichi Iwasawa within the framework of cyclotomic fields. In the early 1970s, Barry Mazur broadened this perspective by applying Iwasawa theory to the Selmer groups of elliptic curves—and more generally, abelian varieties—at good ordinary primes. Later, Perrin-Riou, along with Pollack and Kobayashi, extended these ideas to the realm of supersingular primes. The resulting Iwasawa invariants offer deep insights into the arithmetic properties of these structures. This thesis is dedicated to examining these invariants. In particular, it seeks to establish arithmetic criteria that guarantee the minimal possible values of the Iwasawa invariants. While Gold's criterion addresses this issue in the classical context of Iwasawa theory, our goal is to develop an analogous criterion for elliptic curves of rank one. Furthermore, numerical evidence indicates that for a set of primes of density one, the Iwasawa invariants for rank-one elliptic curves achieve their smallest conceivable values. This work provides additional support for that conjecture.