On the index of the Heegner subgroup of elliptic curves

TL;DR

This paper investigates the index of the subgroup generated by traces of Heegner points on elliptic curves of rank one, proposing a conjecture relating the index to the real locus and Tate-Shafarevich group, supported by extensive computations.

Contribution

It introduces a conjecture linking the index of Heegner point subgroups to geometric and arithmetic properties of elliptic curves, supported by computational evidence.

Findings

Conjecture that I > 1 implies multiple connected components or non-trivial Tate-Shafarevich group.

Computational verification for N < 129999 supports the conjecture.

Provides insights into the structure of rational points generated by Heegner points.

Abstract

Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich group is non-trivial. This conjecture is backed by computations performed on each E that satisfies the above hypothesis in the range N < 129999. This paper was prepared for the proceedings of the Conference on Algorithmic Number Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as possible.