Indivisibility of Heegner points and arithmetic applications

TL;DR

This paper proves a significant equality in the theory of Heegner points, extending previous divisibility results to the full conjecture, using advanced tools like Zhang's proof and Kolyvagin's theorems.

Contribution

It advances the understanding of Heegner points by establishing the predicted equality in Perrin-Riou's main conjecture under more general conditions.

Findings

Proves the equality in Perrin-Riou's Heegner point main conjecture.

Allows for classical Heegner hypothesis and non-squarefree conductors.

Utilizes Zhang's proof of Kolyvagin's conjecture and explicit reciprocity laws.

Abstract

We upgrade Howard's divisibility towards Perrin-Riou's Heegner point main conjecture to the predicted equality. Contrary to previous works in this direction, our main result allows for the classical Heegner hypothesis and non-squarefree conductors. The main ingredients we exploit are W.~Zhang's proof of Kolyvagin's conjecture, Kolyvagin's structure theorem for Shafarevich--Tate groups, and the explicit reciprocity law for Heegner points.