Please use this identifier to cite or link to this item: doi:10.22028/D291-26328
Title: The distribution of group structures on elliptic curves over finite prime fields
Author(s): Gekeler, Ernst-Ulrich
Language: English
Year of Publication: 2006
Free key words: elliptic curves over finite fields
group structures
counting functions
DDC notations: 510 Mathematics
Publikation type: Other
Abstract: 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}).
Link to this record: urn:nbn:de:bsz:291-scidok-46324
Series name: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Series volume: 164
Date of registration: 6-Mar-2012
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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