Simple constructions for elliptic cubic curves with specified small numbers of points
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.
scientific article; zbMATH DE number 4078132
scientific article