On the Mordell-Weil group of elliptic curves induced by families of Diophantine triples

Abstract

The problem of the extendibility of Diophantine triples is closely connected with the Mordell-Weil group of the associated elliptic curve. In this paper, we examine Diophantine triples $\{k-1,k+1,c_l(k)\}$ and prove that the torsion group of the associated curves is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for $l=3,4$ and $l\equiv 1$ or $2 \pmod{4}$. Additionally, we prove that the rank is greater than or equal to 2 for all $l\ge2$. This represents an improvement of previous results by Dujella, Peth\H{o} and Najman, where cases $k=2$ and $l\le3$ were considered.