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 hdl:20.500.11880/26384 http://dx.doi.org/10.22028/D291-26328 |
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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preprint_164_06.pdf | 258,15 kB | Adobe PDF | View/Open |
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