On elliptic curves induced by rational Diophantine quadruples

TL;DR

This paper explores elliptic curves derived from rational Diophantine quadruples, demonstrating the existence of infinitely many such quadruples with specific torsion groups and constructing curves with significant rank.

Contribution

It establishes the infinite occurrence of rational Diophantine quadruples inducing elliptic curves with particular torsion groups and constructs examples with large rank.

Findings

Infinite rational Diophantine quadruples with specified torsion groups

Construction of elliptic curves with large rank in each torsion case

Demonstration of the diversity of elliptic curves from Diophantine quadruples

Abstract

In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/k\mathbb{Z}$ for $k = 2, 4, 6, 8$, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.