An average asymptotic for the number of extremal primes of elliptic curves
Volume 183 / 2018
Abstract
Let $E/\mathbb Q$ be an elliptic curve, and let $p$ be a rational prime of good reduction. Let $a_p(E)$ denote the trace of the Frobenius endomorphism of $E$ at $p$. We say $p$ is a champion prime of $E$ if $\newcommand{\mgi}[1]{[|#1|]}a_p(E)=- \mgi{2\sqrt{p}}$, which occurs precisely when the group of $\mathbb F_p$-rational points is as large as possible in accordance with the Hasse bound. In a similar vein, we say $p$ is a trailing prime of $E$ if $\newcommand{\mgi}[1]{[|#1|]}a_p(E) = +\mgi{2\sqrt{p}}$, which occurs precisely when the group of $\mathbb F_p$-rational points is as small as possible in accordance with the Hasse bound. Together, we say that these primes constitute the extremal primes of $E$. We prove that on average, the number of champion primes of elliptic curves that are less than $X$ is asymptotically equal to $\frac{8}{3\pi} \cdot X^{1/4} /\! \log{X}$. As an immediate corollary, we also gain asymptotics on the average number of trailing primes less than $X$ and the average number of extremal primes less than $X$.
