Geodesic
The distribution of group structures on elliptic curves over finite prime fields
Documenta mathematica, Tome 11 (2006), pp. 119-142.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $p \equiv 1~(\ell^r)$.
Classification : 11, N, 45, G, 20, S, 80
Keywords: elliptic curves over finite fields, group structures, counting functions 
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     author = {Gekeler, Ernst-Ulrich},
     title = {The distribution of group structures on elliptic curves over finite prime fields},
     journal = {Documenta mathematica},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a13/}
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Gekeler, Ernst-Ulrich. The distribution of group structures on elliptic curves over finite prime fields. Documenta mathematica, Tome 11 (2006), pp. 119-142. http://geodesic.mathdoc.fr/item/DOCMA_2006__11__a13/