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The Frequency of Elliptic Curve Groups over Prime Finite Fields

Published online by Cambridge University Press:  20 November 2018

Vorrapan Chandee
Affiliation:
Department of Mathematics, Burapha University, 169 Long-hard Bangsaen rd, Saen suk, Mueang, Chonburi, 20131 Thailand e-mail: vorrapan@buu.ac.th
Chantal David
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montréal, QC, H3G 1M8, Canada e-mail: cdavid@mathstat.concordia.ca
Dimitris Koukoulopoulos
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada e-mail: koukoulo@dms.umontreal.ca
Ethan Smith
Affiliation:
Department of Mathematics, Liberty University, 1971 University Blvd, MSC Box 710052, Lynchburg, VA 24502, USA e-mail: ecsmith13@liberty.edu
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Abstract

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Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${{\mathbb{F}}_{p}}$, we study the frequency to which a given group $G$ arises as a group of points $E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form ${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group ${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when $m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when $m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer $N$ arises as a group order $\#E\left( {{\mathbb{F}}_{p}} \right)$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

References

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