Averages of the number of points on elliptic curves

Abstract

If $E$ is an elliptic curve defined over $\mathbb{Q}$ 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 $\sim K^{\ast }\left(N\right)∕logN$ 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.