High rank elliptic curves induced by rational Diophantine triples

TL;DR

This paper introduces a new method to construct high-rank elliptic curves over Q using rational Diophantine triples, achieving record ranks of 12 and infinite families with rank at least 7.

Contribution

It presents a novel parametrization technique for rational Diophantine triples to generate elliptic curves with high rank, including the current record rank of 12.

Findings

Constructed an elliptic curve with rank 12 induced by a rational Diophantine triple.

Developed an infinite family of elliptic curves with rank >= 7.

Established new record ranks for elliptic curves induced by rational Diophantine triples.

Abstract

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that kind of curves.