On the cyclicity of elliptic curves over finite field extensions
type: Item
instance of: scholarly article
title: On the cyclicity of elliptic curves over finite field extensions
zbMATH Open document ID: 1022.11030
published in: Finite Fields and their Applications
publication date: 2003-05-19T00:00:00Z
review text: Let \(E\) be an elliptic curve over the finite field \(\mathbb{F}_q\) such that \(E(\mathbb{F}_q)\), its group of \(\mathbb{F}_q\)-rational points, is cyclic. Put \(C(E)=\{n: E(\mathbb{F}_{q^n})\) is cyclic\}. The author obtains some general properties of the sets \(C(E)\) and gives a complete description if \(E\) is supersingular. If \(T\) is a positive integer, let \(c(E,T)\) denote the number of elements in \(C(E)\) that are at most \(T\). The author also obtains results on the asymptotic value of \(c(E,T)/T\). In the last section, the author considers the related question of the size of the \(N\)-division field of \(E\) and obtains bounds for this number if \(E\) is supersingular. The problems studied here were raised by \textit{I. E. Shparlinski} in Chapter 6 of his book [Computational and algorithmic problems in finite fields. Kluwer (1992; Zbl 0780.11064)].
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1369826
zbMATH Keywords: cyclic elliptic curve
zbMATH Keywords: supersingular elliptic curve
author: Serge Vlăduţ
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: ffta.1999.0246
OpenAlex ID: W183871989
cites work: Q3139838
cites work: Q3141896
cites work: Nonsingular plane cubic curves over finite fields
cites work: Q5580133
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q4886127
cites work: Q3710637
cites work: Q3997777
DOI: 10.1006/FFTA.1999.0246
description: scientific article; zbMATH DE number 1369826