On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction

TL;DR

This paper provides a concise proof of the anticyclotomic main conjecture for elliptic curves with good ordinary reduction, explicitly relating Fitting ideals of Selmer groups to theta elements.

Contribution

It offers a simplified proof of the anticyclotomic main conjecture and explicitly determines initial Fitting ideals using Bertolini--Darmon theta elements.

Findings

Complete determination of initial Fitting ideals of Selmer groups

Simplified proof of the anticyclotomic main conjecture

Explicit relation to theta elements

Abstract

We give a short proof of the anticyclotomic analogue of the "strong" main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic $\mathbb{Z}_p$-extension in terms of Bertolini--Darmon's theta elements.