On the torsion group of elliptic curves induced by Diophantine triples 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 × Z/10Z, Z/2Z × Z/12Z and Z/4Z × 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.

Elliptic curves induced by Diophantine triples

“…Let us mention that in [8] the authors constructed elliptic curves induced by Diophantine triples with torsion groups Z/2Z × Z/10Z, Z/2Z × Z/12Z and Z/4Z × Z/4Z over quadratic fields.…”

On elliptic curves induced by rational Diophantine quadruples

“…We say that this elliptic curve is induced by the rational Diophantine triple {a, b, c}. The question of possible Mordell-Weil groups of such elliptic curve over Q, Q(t) and quadratic fields, was considered in several papers (see [1,5,8,13,19,20,21,22,23,32]). In particular, it is shown in [8] that all four torsion groups that are allowed by Mazur's theorem for elliptic curves with full 2-torsion, i.e.…”

Rank zero elliptic curves induced by rational Diophantine triples

“…Moreover, it was shown that every elliptic curve with torsion group Z/2Z × Z/8Z is induced by a Diophantine triple (see also [4]). Questions about the ranks of elliptic curves induced by Diophantine triples were studied in several papers ( [1,7,8,10,12,18,19,20,21]). In particular, such curves were used for finding elliptic curves with the largest known rank over Q and Q(t) with torsion groups Z/2Z × Z/4Z ( [18,20]) and Z/2Z × Z/6Z ( [19]).…”