The Distribution of Group Structures on Elliptic Curves over Finite Prime Fields

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 $p \equiv 1~(\ell^r)$.

2000 Mathematics Subject Classification: 11 N 45, 11 G 20, 11 S 80

Keywords and Phrases: Elliptic curves over finite fields, group structures, counting functions

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