engleski

High rank elliptic curves induced by rational Diophantine triples

A rational Diophantine triple is a set of three nonzero rational a, b, c with the property that ab + 1, ac + 1, bc + 1 are perfect squares. We say that the elliptic curve y^2 = (ax + 1)(bx + 1)(cx + 1) is induced by the triple {; ; ; ; ; a, b, c}; ; ; ; ; . In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

Elliptic curves, Diophantine triples, rank

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