Averages of the number of points on elliptic curves

TL;DR

This paper investigates the average behavior of the number of points on elliptic curves over finite fields, focusing on a complex function $K^{ ext{*}}(N)$, and establishes statistical properties and distributions related to it.

Contribution

The paper proves new statistical results about the function $K^{ ext{*}}(N)$, including its mean value and distribution, overcoming challenges posed by its non-multiplicative nature.

Findings

Determined the mean value of $K^{ ext{*}}(N)$ over various sets of $N$.

Established the existence of a distribution function for $K^{ ext{*}}(N)$.

Connected the results to conjectures like Koblitz's on elliptic curve point counts.

Abstract

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. 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 odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an explicitly-given function $K^{\ast}(N)$. The function $K^{\ast}(N)$ 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 $K^{\ast}(N)$. For example, we determine the mean value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that $K^{\ast}(N)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $\# E(\mathbb F_p)$, such as Koblitz's conjecture.