Conjectures Concerning the Orders of the Torsion Subgroup, the Arithmetic Component Groups, and the Cuspidal Subgroup

Abstract

We make several conjectures concerning the relations between the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup of an optimal elliptic curve. These conjectures have implications for the second part of the Birch–Swinnerton-Dyer conjecture.

2000 AMS Subject Classification:

• 11G05
• 11G40

Keywords:

• elliptic curve
• torsion subgroup
• arithmetic component group
• cuspidal subgroup

Acknowledgments

We are very grateful to William Stein for his extensive help on this article. In particular, he provided numerical evidence for many of the conjectures made in this article, and also introduced the author to programming in Sage. Some of the computations were done on sage.math.washington.edu, which is supported by National Science Foundation Grant No. DMS-0821725. The author is also grateful to the Tata Institute of Fundamental Research for a visit during which he first started investigating the issues discussed in this article.

The author was supported by the National Science Foundation under Grant No. 0603668 and by National Security Agency Grant No. Hg8230-10-1-0208. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.