Indivisibility of Heegner cycles over Shimura curves and Selmer groups

TL;DR

This paper demonstrates that Heegner cycles over Shimura curves form a bipartite Euler system, leading to a converse result for the Gross-Zagier-Kolyvagin theorem in higher weight modular forms, linking Selmer groups and Heegner cycles.

Contribution

It establishes the bipartite Euler system structure of Heegner cycles and proves a new converse theorem relating Selmer group rank to Heegner cycle non-vanishing.

Findings

Heegner cycles form a bipartite Euler system over Shimura curves.

A converse to the Gross-Zagier-Kolyvagin theorem for higher weight modular forms is proved.

Non-zero Abel-Jacobi images correspond to residual Selmer group rank one.

Abstract

In this article, we show that the Abel-Jacobi images of the Heegner cycles over the Shimura curves constructed by Nekovar, Besser and the theta elements contructed by Chida-Hsieh form a bipartite Euler system in the sense of Howard. As an application of this, we deduce a converse to Gross-Zagier-Kolyvagin type theorem for higher weight modular forms generalizing works of Wei Zhang and Skinner for modular forms of weight two. That is, we show that if the rank of certain residual Selmer group is one, then the Abel-Jacobi image of the Heegner cycle is non-zero in the residual Selmer group.