On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

TL;DR

This paper investigates the torsion groups of elliptic curves derived from Diophantine triples over quadratic fields, demonstrating that certain torsion groups not present over rationals do occur infinitely often over quadratic fields.

Contribution

It proves the existence of elliptic curves with specific torsion groups induced by Diophantine triples over quadratic fields, including infinitely many such triples.

Findings

All three torsion groups appear over some quadratic field.

Infinitely many Diophantine triples induce elliptic curves with these torsion groups.

Certain torsion groups not over Q are realized over quadratic fields.

Abstract

The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z x Z/10Z, Z/2Z x Z/12Z and Z/4Z x Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.