A Cohen–Lenstra phenomenon for elliptic curves

Given an elliptic curve E and a finite Abelian group G, we consider the problem of counting the number of primes p for which the group of points modulo p is isomorphic to G. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves.

[1]  M. Deuring.  Die Typen der Multiplikatorenringe elliptischer Funktionenkörper , 1941 .

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

[3]  D. A. Burgess.  On Character Sums and L-Series. II , 1962 .

[4]  H. Davenport.  Multiplicative Number Theory , 1967 .

[5]  H. Trotter,et al.  Frobenius Distributions in GL2-Extensions: Distribution of Frobenius Automorphisms in GL2-Extensions of the Rational Numbers , 1976 .

[6]  Henri Cohen,et al.  Heuristics on class groups , 1984 .

[7]  Henri Cohen,et al.  Heuristics on class groups of number fields , 1984 .

[8]  H. W. Lenstra,et al.  Factoring integers with elliptic curves , 1987 .

[9]  René Schoof,et al.  Nonsingular plane cubic curves over finite fields , 1987, J. Comb. Theory A.

[10]  N. Koblitz.  PRIMALITY OF THE NUMBER OF POINTS ON AN ELLIPTIC CURVE OVER A FINITE FIELD , 1988 .

[11]  N. Elkies.  Distribution of supersingular primes , 1991 .

[12]  Kevin James Clemson,et al.  Average Frobenius distribution of elliptic curves , 1998 .

[13]  F. Pappalardi.  AVERAGE FROBENIUS DISTRIBUTION FOR INERTS IN Q(i) , 2003 .

[14]  H. Iwaniec,et al.  Analytic Number Theory , 2004 .

[15]  K. James.  Average frobenius distributions for elliptic curves with 3-torsion , 2004 .

[16]  Francesco Pappalardi,et al.  Average Frobenius distribution for inerts in $Bbb Q(i)$ , 2004 .

[17]  Glyn Harman,et al.  ANALYTIC NUMBER THEORY (American Mathematical Society Colloquium Publications 53) , 2005 .

[18]  Emmanuel Kowalski,et al.  Analytic problems for elliptic curves , 2005, math/0510197.

[19]  D. Ivanov,et al.  Average Frobenius distributions for elliptic curves with nontrivial rational torsion , 2005 .

[20]  Hugh L. Montgomery,et al.  Multiplicative Number Theory I: Classical Theory , 2006 .

[21]  A. Balog,et al.  Average twin prime conjecture for elliptic curves , 2007 .

[22]  Christopher J. Hillar,et al.  Automorphisms of Finite Abelian Groups , 2006, Am. Math. Mon..

[23]  Alessandro Languasco,et al.  ON THE MONTGOMERY–HOOLEY THEOREM IN SHORT INTERVALS , 2010 .

[24]  Neil J. Calkin,et al.  Average Frobenius distributions for elliptic curves over abelian extensions , 2011 .

[25]  K. James,et al.  Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals , 2011, Mathematical Proceedings of the Cambridge Philosophical Society.

[26]  Averages of the number of points on elliptic curves , 2012, 1208.0919.

[27]  Ethan Smith,et al.  Elliptic curves with a given number of points over finite fields , 2011, Compositio Mathematica.

[28]  Igor E. Shparlinski,et al.  On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields , 2010, Exp. Math..