Rank zero elliptic curves induced by rational Diophantine triples

TL;DR

This paper investigates the minimal possible rank of elliptic curves induced by rational Diophantine triples, providing the first known example with positive elements that induce rank zero elliptic curves.

Contribution

It presents the first known example of a rational Diophantine triple with positive elements inducing a rank zero elliptic curve.

Findings

Existence of rational Diophantine triples with mixed signs inducing rank 0.

First known example of positive-element Diophantine triple inducing rank 0.

Insights into the minimal rank achievable by such elliptic curves.

Abstract

Rational Diophantine triples, i.e. rationals a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.