On some elliptic curves defined over Q of free rank ?9

Abstract

By Q (respectively Z ) we denote the field of rational numbers (respectively the ring of rational integers). The famous theorem of Mordell states that the set of all Qrational points on an elliptic curve defined over Q is a finitely generated abelian group. The rank of such a curve is defined to be the number of free generators of this group. Wiman [9] had examples of rank ~ 4 elliptic curves. D. Penny and C. Pomerance [5], [6] found examples of elliptic curves of rank ~ 7 by using computer. F. G. Grunewald and R. Zimmert [i] constructed examples of elliptic curves of rank ~ 8. Although N~ron [3], [4] proved that elliptic curves of rank ~ ii exist, there are no examples. It seems to be a practical benefit that we have examples of elliptic curves with large rank. In this paper, we shall explicitly construct an infinity of elliptic curves of rank ~ 9, modifying the method in [i].