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## Acta Arithmetica

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## An average asymptotic for the number of extremal primes of elliptic curves

### Volume 183 / 2018

Acta Arithmetica 183 (2018), 145-165 MSC: Primary 11G05; Secondary 11F30. DOI: 10.4064/aa170406-30-1 Published online: 30 March 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 $X$.

#### Authors

• Luke GibersonDepartment of Mathematical Sciences
Clemson University
O-110 Martin Hall, Box 340975
Clemson, SC 29634, U.S.A.
e-mail
• Kevin JamesDepartment of Mathematical Sciences
Clemson University
O-110 Martin Hall, Box 340975
Clemson, SC 29634, U.S.A.
e-mail

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