Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

TL;DR

This paper advances the understanding of the two-variable main conjecture in Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields by constructing p-adic L-functions, analyzing Selmer groups, and exploring implications of base change.

Contribution

It introduces new results on the existence of p-adic L-functions, the structure of dual Selmer groups, and criteria for divisibility in the main conjecture, extending prior work in Iwasawa theory.

Findings

Existence of a bounded p-adic L-function for elliptic curves over imaginary quadratic fields.

A formula for the corank of the p-primary Tate-Shafarevich group in Z_p^2-extensions.

A criterion for divisibility of the main conjecture via specializations to cyclotomic characters.

Abstract

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z_p^2-extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.