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 Z/2Z × Z/2Z for l = 3, 4 and l ≡ 1 or 2 (mod 4). Additionally, we prove that the rank is greater than or equal to 2 for all l ≥ 2. This represents an improvement of previous results by Dujella, Pethő and Najman, where cases k = 2 and l ≤ 3 were considered. 2010 AMS Mathematics subject classification. Primary 11G05. Keywords and phrases. Diophantine triples, elliptic curves, Mordell-Weil group.

There Are Infinitely Many Rational Diophantine Sextuples

“…It is an open problem whether this torsion group is possible for elliptic curves induced by an integer Diophantine triple (see e.g. [13,24]). On the other hand, examples of elliptic curves, induced by rational Diophantine triples, with torsion group Z/2Z × Z/6Z and rank equal to 1, 2, 3 and 4 can be found in [9] (for examples of elliptic curves with torsion groups Z/2Z×Z/4Z and Z/2Z×Z/8Z…”