Iwasawa Main Conjecture for Heegner Points: Supersingular Case

TL;DR

This paper proves an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with a_p=0, advancing understanding of the arithmetic of these curves in the supersingular case.

Contribution

It establishes the Iwasawa main conjecture in the supersingular case with a_p=0, incorporating a novel approach for the '$$' sign setting.

Findings

Proves the anticyclotomic Iwasawa main conjecture for supersingular elliptic curves with a_p=0.

Improves Skinner's result on the converse of the Gross--Zagier--Kolyvagin theorem.

Utilizes recent divisibility results and Howard's argument adapted to the '$$' setting.

Abstract

In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the second author on one divisibility in the Iwasawa--Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions, together with an argument of Howard (adapted to our "$\pm$"-situation). As a byproduct, we also obtain an improvement of Skinner's result on a converse to the Gross--Zagier--Kolyvagin theorem.