Izvorni znanstveni članak

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

High rank elliptic curves induced by rational Diophantine triples

Andrej Dujella orcid.org/0000-0001-6867-5811 ; Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
Juan Carlos Peral ; Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain

Sažetak

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 $\mathbb{Q}$ 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 $\geq 7$, which are both the current records for that kind of curves.

Ključne riječi

Elliptic curves; Diophantine triples; rank

Hrčak ID:

248665

URI

https://hrcak.srce.hr/248665

Datum izdavanja:

23.12.2020.

Posjeta: 1.230 *