Symmetric diophantine systems and families of elliptic curves of high rank

Abstract

While there has been considerable interest in the problem of finding elliptic curves of high rank over Q, very few parametrized families of elliptic curves of generic rank ≥ 8 have been published. In this paper we use solutions of certain symmetric diophantine systems to construct several parametrized families of elliptic curves with their generic ranks ranging from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve ≤ 13. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves with much higher rank.Mathematics Subject Classification: 11G05, 11D25, 11D41