Fine Selmer groups of elliptic curves over p-adic Lie extensions

Abstract

The aim of this paper is to discuss variants for elliptic curves of some deep conjectures of classical cyclotomic Iwasawa theory. Let F be a finite extension of Q, p an odd prime number, and F cyc the cyclotomic Zp-extension of Q. Let K(F cyc) denote the maximal unramified abelian p-extension of F cyc, in which every prime of F cyc above p splits completely, and define Y (F cyc) to be the Galois group of K(F cyc) over F cyc. Put = G(F cyc/F ), and write ( ) for the Iwasawa algebra of . Iwasawa proved that Y (F cyc) is always a finitely generated torsion ( )-module, and he conjectured [12] that in fact Y (F cyc) is always a finitely generated Zp-module. At present, this conjecture has only been proven when F is abelian over Q (see [5],[26]). Perhaps surprisingly, it does not seem to have been pointed out in the literature that there is a precise analogue of Iwasawa’s conjecture for elliptic curves over F cyc. Let E be an elliptic curve over F , and let