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

Abstract

We determine the probability that a randomly chosen elliptic curve $E/{\F}_p$ over a randomly chosen prime field ${\F}_p$ has an ${\ell}$-primary part $E({\F}_p) [\ell^{\infty}]$ isomorphic with a fixed abelian $\ell$-group $H^{(\ell)}_{\alpha,\beta} = {\Z}/{\ell}^{\alpha} \times {\Z}/\ell^{\beta}. \smallskip$Probabilities for "$|E(\F_p)|$ divisible by $n'', ``E(\F_p)$ cyclic" and expectations for the number of elements of precise order $n$ in $E(\F_p)$ are derived, both for unbiased $E/\F_p$ and for $E/\F_p$ with $p \equiv 1~(\ell^r)$.