Construction d’une courbe elliptique de rang \(\geq 12\). (English) Zbl 0541.14027

Mestre, Jean-François

C. R. Acad. Sci., Paris, Sér. I 295, 643-644 (1982).

The author describes a method for constructing elliptic curves E of high rank defined over the rational numbers. The basic idea is that the rank of E is likely to be large if the number \(N_ p\) of points of E modulo p (where p is a prime at which E has good reduction) is as large as possible for all \(p\leq P_ 0,\) where \(P_ 0\) will depend on the computing capacity availabe. Taking \(P_ 0=17\), the author obtains various curves of ranks 6,7,8,9; taking \(P_ 0=37\), he obtains a curve of rank at least 12.

Reviewer: I.G.Macdonald

MSC:

14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves
14H25	Arithmetic ground fields for curves
14-04	Software, source code, etc. for problems pertaining to algebraic geometry

Keywords:

constructing elliptic curves of high rank