engleski

Mordell-Weil groups of elliptic curves induced by Diophantine triples

We study the possible structure of the groups of rational points on elliptic curves of the form y^2 = (x + ab)(x + ac)(x + bc), where a, b, c are nonzero rationals such that the product of any two of them is one less than a square. Such a triple {; ; a, b, c}; ; is called a rational Diophantine triple. There are exactly four types of possible torsion groups for elliptic curves of this form. In each case, we construct examples and parametric families of elliptic curves with relatively high rank. In particular, we describe a joint work with Juan Carlos Peral, with construction of an elliptic curve over the field of rational functions Q(t) with torsion group Z/2Z x  Z/4Z and generic rank equal to 4, and an elliptic curve over Q with the same torsion group and rank 9. Both results improve previous records for ranks of curves with this torsion group.

elliptic curves ; torsion group ; rank ; Diophantine triples

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