Elliptic curves with a given number of points over finite fields

TL;DR

This paper investigates the distribution of primes for which elliptic curves have a specific number of points, providing improved average bounds and asymptotic formulas under certain hypotheses.

Contribution

It introduces new bounds on the average number of such primes and derives an asymptotic formula under conjectural assumptions.

Findings

Better average bounds than the Hasse bound

Asymptotic formula for prime counts under hypotheses

Conditional results based on prime distribution conjectures

Abstract

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.