Averages of the number of points on elliptic curves

Martin, Greg; Pollack, Paul; Smith, Ethan

doi:10.2140/ant.2014.8.813

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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

DOI: 10.2140/ant.2014.8.813

Abstract

If $E$ is an elliptic curve defined over $ℚ$ and $p$ is a prime of good reduction for $E$ , let $E (F_{p})$ denote the set of points on the reduced curve modulo $p$ . Define an arithmetic function $M_{E} (N)$ by setting $M_{E} (N) : = # {p : # E (F_{p}) = N}$ . Recently, David and the third author studied the average of $M_{E} (N)$ 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 $M_{E} (N)$ over a box with sufficiently large sides is $\sim K^{*} (N) ∕ log N$ 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 $# E (F_{p})$ , such as Koblitz’s conjecture.

Keywords

elliptic curves, Koblitz conjecture, mean values of arithmetic functions

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11N37, 11N60

Milestones

Received: 26 August 2012

Revised: 14 December 2013

Accepted: 15 February 2014

Published: 10 August 2014

Authors

Greg Martin
Department of Mathematics University of British Columbia Room 121 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada

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

Vol. 8, No. 4, 2014