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

### Ernst-Ulrich Gekeler

FR 6.1 Mathematik Universität des Saarlandes D-66041 Saarbrücken Germany

## Abstract

We determine the probability that a randomly chosen elliptic curve $E/F_{p}$ over a randomly chosen prime field $F_{p}$ has an $ℓ$-primary part $E(F_{p})[ℓ_{∞}]$ isomorphic with a fixed abelian $ℓ$-group $H_{α,β}=Z/ℓ_{α}×Z/ℓ_{β}$.

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 $p≡1(ℓ_{r})$.

## Cite this article

Ernst-Ulrich Gekeler, The distribution of group structures on elliptic curves over finite prime fields. Doc. Math. 11 (2006), pp. 119–142

DOI 10.4171/DM/206