On the Iwasawa theory of CM fields for supersingular primes

TL;DR

This paper proves new main conjectures in Iwasawa theory for CM fields and elliptic curves, removing previous restrictions and connecting to the Birch and Swinnerton-Dyer conjecture.

Contribution

It establishes two-variable main conjectures for CM fields and plus/minus main conjectures for CM elliptic curves without the ordinary hypothesis, using Rubin-Stark Kolyvagin systems.

Findings

Proved a two-variable main conjecture for CM fields.

Established the Park-Shahabi plus/minus main conjecture for CM elliptic curves.

Derived implications for the Birch and Swinnerton-Dyer conjecture.

Abstract

The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $F$ without assuming the $p$-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark $\mathcal{L}$-restricted Kolyvagin systems which is constructed out of the conjectural Rubin-Stark Euler system of rank $g$. (We are also able to obtain weaker unconditional results in this direction.) Second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve $E$ defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for $E$.