Elliptic Curves of High Rank with Nontrivial Torsion Group over Q

In order to find elliptic curves over Q with large rank, J.-F. Mestre [1991] constructed an infinite family of elliptic curves with rank at least 12. Then K. Nagao [1994] and S. Kihara [1997b] found infinite subfamilies of rank 13 and 14, respectively. By specialization, elliptic curves of rank at least 21 [Nagao and Kouya 1994], 22 [Fermigier 1997], and 23 [Martin and McMillen 1997] have also been found. There have been quite a few efforts to construct families of curves with large rank having a nontrivial prescribed torsion group [Fermigier 1996; Nagao 1997; Kihara 1997c; 1997a; Kulesz 1998; 1999]. For example, S. Fermigier constructed a family of elliptic curves with rank at least 8 and a nontrivial point of order 2 and specialized it to a curve of rank 14. Here we improve the rank records for curves with torsion group Z/3Z, Z/4Z, Z/5Z, Z/6Z, Z/7Z, Z/8Z, and Z/2Z x Z/2Z.