Anticyclotomic Iwasawa theory of CM elliptic curves

TL;DR

This paper investigates the Iwasawa theory of CM elliptic curves over anticyclotomic extensions, establishing the structure of Selmer groups and proving a main conjecture in cases of odd order vanishing of the L-function.

Contribution

It demonstrates that for odd order vanishing of the L-function, the dual Selmer group has rank one and proves a form of the Iwasawa main conjecture for its torsion part.

Findings

Dual Selmer group has rank exactly one in odd vanishing cases.

Main conjecture holds for the torsion submodule of the Selmer group.

Results connect the order of vanishing of L-functions with Selmer group structure.

Abstract

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic $\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, the two variable main conjecture of Rubin implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg shows that it is not a torsion module. In this paper we show that in the case of odd order of vanishing the dual of the Selmer group has rank exactly one, and we prove a form of the Iwasawa main conjecture for the torsion submodule.