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2019 Families of elliptic curves of rank $\geq 5$ over $\mathbb Q(t)$
Dustin Moody, Mohammad Sadek, Arman Shamsi Zargar
Rocky Mountain J. Math. 49(7): 2253-2266 (2019). DOI: 10.1216/RMJ-2019-49-7-2253

Abstract

We construct two infinite families of curves with high rank. The first is defined by the equation $y^2=x(x-a^2)(x-b^2)+a^2b^2$, where $a,b \in \mathbb {Q}(t)$. The second family arises from a system of rational cuboids, i.e., a rectangular box for which the lengths of the edges and face diagonals are all rational. We create a second family with defining equation $y^2=(x-a^2)(x-b^2)(x-c^2)+a^2b^2c^2$, where $a,b,c \in \mathbb {Q}(t)$ are the edge lengths of a rational cuboid. We show that the rank of both families is $\geq 5$ over $\mathbb Q(t)$. We conclude by studying corresponding families of curves defined by other known solutions to the rational cuboid problem, and find some specific examples of curves from these various families with high rank.

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Dustin Moody. Mohammad Sadek. Arman Shamsi Zargar. "Families of elliptic curves of rank $\geq 5$ over $\mathbb Q(t)$." Rocky Mountain J. Math. 49 (7) 2253 - 2266, 2019. https://doi.org/10.1216/RMJ-2019-49-7-2253

Information

Published: 2019
First available in Project Euclid: 8 December 2019

zbMATH: 07152863
MathSciNet: MR4039968
Digital Object Identifier: 10.1216/RMJ-2019-49-7-2253

Subjects:
Primary: 11G05

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 7 • 2019
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