Vol. 8, No. 4, 2014

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Averages of the number of points on elliptic curves

Greg Martin, Paul Pollack and Ethan Smith

Vol. 8 (2014), No. 4, 813–836

If E is an elliptic curve defined over and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N) := #{p : #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) over certain “boxes” of elliptic curves E. Assuming a plausible conjecture about primes in short intervals, they showed the following: for each N, the average of ME(N) over a box with sufficiently large sides is K(N)logN for an explicitly given function K(N).

The function K(N) is somewhat peculiar: defined as a product over the primes dividing N, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about K(N). For example, we determine the mean value of K(N) over all N, odd N and prime N, and we show that K(N) has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of #E(Fp), such as Koblitz’s conjecture.

elliptic curves, Koblitz conjecture, mean values of arithmetic functions
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11N37, 11N60
Received: 26 August 2012
Revised: 14 December 2013
Accepted: 15 February 2014
Published: 10 August 2014
Greg Martin
Department of Mathematics
University of British Columbia
Room 121
1984 Mathematics Road
Vancouver, BC V6T 1Z2
Paul Pollack
Department of Mathematics
University of Georgia
Boyd Graduate Studies Research Center
Athens, GA 30602
United States
Ethan Smith
Department of Mathematics
Liberty University
1971 University Blvd.
Lynchburg, VA 24502
United States