The frequency of elliptic curve groups over prime finite fields

Abstract

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${\mathbb{F}}_p$, we study the frequency to which a given group $G$ arises as a group of points $E({\mathbb{F}}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon }$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E({\mathbb{F}}_p)$.

40 pages, with an appendix by Chantal David, Greg Martin and Ethan Smith. Final version, to appear in the Canad. J. Math. Major reorganization of the paper, with the addition of a new section, where the main results are summarized and explained

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