Simple constructions for elliptic cubic curves with specified small numbers of points
type: Item
instance of: scholarly article
title: Simple constructions for elliptic cubic curves with specified small numbers of points
zbMATH Open document ID: 0659.51012
DOI: 10.1016/S0195-6698(88)80004-0
author: Anthony Donald Keedwell
published in: European Journal of Combinatorics
publication date: 1988-01-01T00:00:00Z
review text: In a desarguesian plane of order \(q=p^ n\), the number of points which an elliptic cubic curve can contain is bounded above and below by the Hasse-Weil bounds \(q+1\pm [2\sqrt{q}]\). It is well known [cf. the survey paper by \textit{J. W. P. Hirschfeld}, Lect. Notes Math. 1036, 6-23 (1983; Zbl 0522.51012)] that, with certain exceptions, all numbers of points permitted by the Hasse-Weil bounds can be attained. All the known proofs require extensive use of the general theory of abelian varieties. The purpose of the paper under review is to present very simple actual constructions, without the use of the general theory of abelian varieties or of a computer, for the equation of a curve which is required to have exactly s points in the plane. These constructions are presented for the cases \(s=4,5,6,7,9\), and 11. As a byproduct, the author shows also that an elliptic cubic curves having an odd multiple of 7 points cannot exist in a plane of order \(2^ r\), r odd, and that such a curve having an odd multiple of 11 points cannot exist in a plane of order \(2^ r\) if r is relatively prime to 5.
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 4078132
zbMATH Keywords: desarguesian plane
zbMATH Keywords: elliptic cubic curve
zbMATH Keywords: Hasse-Weil bounds
MaRDI profile type: Publication
cites work: The number of points on an elliptic cubic curve over a finite field
cites work: Q3851217
cites work: Q3672590
cites work: Nonsingular plane cubic curves over finite fields
cites work: On the number of points of elliptic curves over a finite field and a problem of B. Segre
cites work: Abelian varieties over finite fields
full work available at URL: s0195-6698(88)80004-0
OpenAlex ID: W2002654821
Recommended article: Q4276193
Recommended article: Q4859142
Recommended article: Cubic curves, finite geometry and cryptography
Recommended article: Q5730228
Recommended article: On Equations y2=xn+k in a Finite Field
Recommended article: Q4906689
Recommended article: On a method of solving Diophantine equations
Recommended article: Q3717206
Recommended article: Q3788126
Recommended article: Un analogue du calcul de Heegner. (An analogue of the Heegner calculus)
description: scientific article; zbMATH DE number 4078132