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High-rank elliptic curves with torsion $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ induced by Diophantine triples

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 June 2014

Andrej Dujella  and
Juan Carlos Peral
Andrej Dujella
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia email duje@math.hr
Juan Carlos Peral
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain email juancarlos.peral@ehu.es

Abstract

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We construct an elliptic curve over the field of rational functions with torsion group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and rank equal to four, and an elliptic curve over $\mathbb{Q}$ with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

MSC classification

Secondary: 11G05: Elliptic curves over global fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 282 - 288
Copyright
© The Author(s) 2014 

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