TY  - GEN
T1  - The distribution of group structures on elliptic curves over finite prime fields
A1  - Gekeler,Ernst-Ulrich
Y1  - 2012/03/06
N2  - 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}^{(l)}=\mathbb{Z}/\ell^{\alpha}\times\mathbb{Z}/\ell^{\beta}.
We show that the probability agrees with the one predicted by a natural though unproven equidistribution hypothesis for Frobenius elements in GL(2,\mathbb{Z}_{\ell}).
Probabilities for "\left|E(\mathbb{F}_{p})\right| 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\equiv1(\ell^{r}). 
CY  - Saarbrücken
PB  - Universitäts- und Landesbibliothek
AD  - Postfach 151141, 66041 Saarbrücken
UR  - http://scidok.sulb.uni-saarland.de/volltexte/2012/4632
ER  -