Heegner points on Hijikata-Pizer-Shemanske curves

TL;DR

This paper investigates the construction of Heegner points on elliptic curves and modular abelian varieties via Shimura curves associated with quaternionic orders, addressing BSD conjecture predictions.

Contribution

It extends the theory of Heegner points to a broad class of Shimura curves linked to quaternionic orders, confirming BSD conjecture predictions under mild conditions.

Findings

Existence of non-torsion Heegner points in predicted cases

Generalization to Shimura curves from Hijikata-Pizer-Shemanske orders

Support for BSD conjecture in this broader context

Abstract

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic or- ders closely related to those introduced by Hijikata{Pizer{Shemanske in the 80's. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In par- ticular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.