Elliptic curves induced by Diophantine triples

TL;DR

This paper investigates elliptic curves derived from Diophantine triples over Q(t), focusing on their torsion groups and rank, and presents improved results on their rank over Q(t).

Contribution

It provides new insights and improved bounds on the rank of elliptic curves induced by Diophantine triples over Q(t).

Findings

Elliptic curves can have specific torsion groups as classified by Mazur.

The paper improves known results on the rank of these curves over Q(t).

Results contribute to understanding the structure and rank of elliptic curves from Diophantine triples.

Abstract

Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e. Z/2Z x Z/2Z, Z/2Z x Z/4Z, Z/2Z x Z/6Z or Z/2Z x Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.