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 $H_{\alpha ,\beta }^{(\ell )}=\mathbb{Z}/{\ell }^{\alpha }\times \mathbb{Z}/{\ell }^{\beta }$.

Probabilities for “$|E({\mathbb{F}}_p)|$ 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 $p\equiv 1({\ell }^r)$.