Algebra & Number Theory

Averages of the number of points on elliptic curves

Greg Martin, Paul Pollack, and Ethan Smith

Full-text: Open access

Abstract

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.

Article information

Source
Algebra Number Theory, Volume 8, Number 4 (2014), 813-836.

Dates
Received: 26 August 2012
Revised: 14 December 2013
Accepted: 15 February 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730195

Digital Object Identifier
doi:10.2140/ant.2014.8.813

Mathematical Reviews number (MathSciNet)
MR3248986

Zentralblatt MATH identifier
1316.11046

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11N37: Asymptotic results on arithmetic functions 11N60: Distribution functions associated with additive and positive multiplicative functions

Keywords
elliptic curves Koblitz conjecture mean values of arithmetic functions

Citation

Martin, Greg; Pollack, Paul; Smith, Ethan. Averages of the number of points on elliptic curves. Algebra Number Theory 8 (2014), no. 4, 813--836. doi:10.2140/ant.2014.8.813. https://projecteuclid.org/euclid.ant/1513730195


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