The distribution of group structures on elliptic curves over finite prime fields

Abstract

We determine the probability that a randomly chosen elliptic curve E/\mathbb{F}_{p} over a randomly chosen prime field \mathbb{F}_{p} has an \ell-primary part E(\mathbb{F}_{p})[\ell^{\infty}] isomorphic with a fixed abelian \ell-group H_{\alpha,\beta}^{(l)}=\mathbb{Z}/\ell^{\alpha}\times\mathbb{Z}/\ell^{\beta}. We show that the probability agrees with the one predicted by a natural though unproven equidistribution hypothesis for Frobenius elements in GL(2,\mathbb{Z}_{\ell}). Probabilities for "\left|E(\mathbb{F}_{p})\right| divisible by n", "E(\mathbb{F}_{p}) cyclic" and expectations for the number of elements of precise order n in E(\mathbb{F}_{p}) are derived, both for unbiased E/\mathbb{F}_{p} and for E/\mathbb{F}_{p} with p\equiv1(\ell^{r}).