On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields

Abstract

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection that correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory; some are conditional under certain widely believed conjectures; and others are purely heuristic in nature.

2000 AMS Subject Classification:

• Primary 11D45
• Secondary 11P05
• 14G05

Keywords:

• elliptic curve
• finite field
• group structure

ACKNOWLEDGMENTS

The authors are grateful to Roger Baker for sending us a preliminary version of his work [Baker 10], to Karl Dilcher for pointing out the relevance of the results of [Ljunggren 43] to this work, to Andrzej Schinzel for a discussion concerning Cramér's conjecture for arithmetic progressions, and to Corrado Falcolini for his help with Mathematica plotting.

The second author was partially supported by GNSAGA from INDAM. The third author was supported in part by ARC Grant DP0881473, Australia, and by NRF Grant CRP2-2007-03, Singapore.