Journal Article

Group Structures of Elliptic Curves Over Finite Fields

Vorrapan Chandee, Chantal David, Dimitris Koukoulopoulos and Ethan Smith

in International Mathematics Research Notices

Volume 2014, issue 19, pages 5230-5248
Published in print January 2014 | ISSN: 1073-7928
Published online June 2013 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1093/imrn/rnt120
Group Structures of Elliptic Curves Over Finite Fields

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It is well known that if E is an elliptic curve over the finite field , then for some positive integers m,k. Let S(M,K) denote the set of pairs (m,k) with mM and kK for which there exists an elliptic curve over some prime finite field whose group of points is isomorphic to . Banks, Pappalardi, and Shparlinski recently conjectured that if , then a density zero proportion of the groups in question actually arises as the group of points on some elliptic curve over some prime finite field. On the other hand, if , they conjectured that a density 1 proportion of the groups in question arises as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range , and we prove that the second part of their conjecture holds in the limited range KM4+ϵ. In the wider range KM2, we show that at least a positive density of the groups in question actually occurs.

Journal Article.  3135 words.  Illustrated.

Subjects: Mathematics

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