Glasnik Matematicki, Vol. 55, No. 2 (2020), 237-252.

HIGH RANK ELLIPTIC CURVES INDUCED BY RATIONAL DIOPHANTINE TRIPLES

Andrej Dujella and Juan Carlos Peral

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: duje@math.hr

Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain
e-mail: juancarlos.peral@ehu.es

Dedicated to the memory of our friend and coauthor Julián Aguirre

Abstract.   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 y2 = (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.

2010 Mathematics Subject Classification.   11G05, 11D09

Key words and phrases.   Elliptic curves, Diophantine triples, rank

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.2.05

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