Glasnik matematički, Vol. 55 No. 2, 2020.
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
Datum izdavanja:
23.12.2020.
Posjeta: 858 *