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2014 Averages of the number of points on elliptic curves
Greg Martin, Paul Pollack, Ethan Smith
Algebra Number Theory 8(4): 813-836 (2014). 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 $E\left({\mathbb{F}}_{p}\right)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function ${M}_{E}\left(N\right)$ by setting ${M}_{E}\left(N\right):=#\left\{p:#E\left({\mathbb{F}}_{p}\right)=N\right\}$. Recently, David and the third author studied the average of ${M}_{E}\left(N\right)$ 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 ${M}_{E}\left(N\right)$ over a box with sufficiently large sides is $\sim {K}^{\ast }\left(N\right)∕\phantom{\rule{0.3em}{0ex}}logN$ for an explicitly given function ${K}^{\ast }\left(N\right)$.

The function ${K}^{\ast }\left(N\right)$ 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 ${K}^{\ast }\left(N\right)$. For example, we determine the mean value of ${K}^{\ast }\left(N\right)$ over all $N$, odd $N$ and prime $N$, and we show that ${K}^{\ast }\left(N\right)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $#E\left({\mathbb{F}}_{p}\right)$, such as Koblitz’s conjecture.

## Citation

Greg Martin. Paul Pollack. Ethan Smith. "Averages of the number of points on elliptic curves." Algebra Number Theory 8 (4) 813 - 836, 2014. https://doi.org/10.2140/ant.2014.8.813

## Information

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

zbMATH: 1316.11046
MathSciNet: MR3248986
Digital Object Identifier: 10.2140/ant.2014.8.813

Subjects:
Primary: 11G05
Secondary: 11N37, 11N60