High rank 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.

arXiv comment11 pages

References in corpus (10)

• There are infinitely many rational Diophantine sextuples
• On Mordell-Weil groups of elliptic curves induced by Diophantine triples
• Three lectures on elliptic surfaces and curves of high rank
• More on Diophantine sextuples
• Diagonal genus 5 curves, elliptic curves over $\mathbb{Q}(t)$, and rational diophantine quintuples
• There are infinitely many rational Diophantine sextuples with square denominators
• Diophantine triples and construction of high-rank elliptic curves
• Rational Diophantine sextuples containing two regular quadruples and one regular quintuple
• Elliptic curves induced by Diophantine triples
• Rank zero elliptic curves induced by rational Diophantine triples

Cited by in corpus (1)

• On elliptic curves induced by rational Diophantine quadruples