On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields

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.

[1]  W. Sierpinski,et al.  Sur certaines hypothèses concernant les nombres premiers , 1958 .

[2]  S. Lang,et al.  Abelian varieties over finite fields , 2005 .

[3]  Andrzej Schinzel,et al.  On the equation $y^m = P(x)$ , 1976 .

[4]  K. Soundararajan.  THE DISTRIBUTION OF PRIME NUMBERS , 2006, math/0606408.

[5]  L. Washington.  Elliptic Curves: Number Theory and Cryptography, Second Edition , 2008 .

[6]  J. Voloch.  A note on elliptic curves over finite fields , 1988 .

[7]  R. Horn,et al.  A heuristic asymptotic formula concerning the distribution of prime numbers , 1962 .

[8]  Wilhelm Ljunggren,et al.  Einige Bemerkungen Über Die Darstellung Ganzer Zahlen Durch Binäre Kubische Formen Mit Positiver Diskriminante , 1942 .

[9]  Y. Bugeaud.  Bounds for the solutions of superelliptic equations , 1997, Compositio Mathematica.

[10]  Everett W. Howe.  On the group orders of elliptic curves over finite fields , 2001, math/0110262.

[11]  L. Washington.  Elliptic Curves: Number Theory and Cryptography , 2003 .

[12]  Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli , 2006, math/0602116.

[13]  Primes in arithmetic progressions to spaced moduli , 2005 .

[14]  R. Sitaramachandrarao.  On an error term of Landau-II , 1985 .

[15]  A. Harles.  Sieve Methods , 2001 .

[16]  Igor E. Shparlinski,et al.  On Group Structures Realized by Elliptic Curves over a Fixed Finite Field , 2010, Exp. Math..

[17]  Michael Rosen.  Abelian Varieties over ℂ , 1986 .

[18]  Tanja Lange,et al.  Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .

[19]  Andrew Granville,et al.  HARALD CRAM ER AND THE DISTRIBUTION OF PRIME NUMBERS , 1993 .

[20]  W. Waterhouse,et al.  Abelian varieties over finite fields , 1969 .

[21]  AN IMPROVEMENT FOR THE LARGE SIEVE FOR SQUARE MODULI , 2005, math/0512271.

[22]  Hans-Georg Rück.  A note on elliptic curves over finite fields , 1987 .

[23]  Y. Bugeaud.  Sur la distance entre deux puissances pures , 1996 .