Symmetric diophantine systems and families of elliptic curves of high rank

TL;DR

This paper constructs parametrized families of elliptic curves with high generic rank (8 to 12) using solutions of symmetric diophantine systems, advancing the understanding of high-rank elliptic curves over rationals.

Contribution

It introduces new parametrized families of elliptic curves with high generic rank, expanding the known examples and methods for constructing such curves.

Findings

Families with generic rank ≥8 to ≥12 constructed

Specific examples with rank ≤13 identified

Potential for even higher rank elliptic curves suggested

Abstract

While there has been considerable interest in the problem of finding elliptic curves of high rank over $\mathbb{Q}$, very few parametrized families of elliptic curves of generic rank $\geq 8$ 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 $\leq 13$. 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.