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2017 Heron quadrilaterals via elliptic curves
Farzali Izadi, Foad Khoshnam, Dustin Moody
Rocky Mountain J. Math. 47(4): 1227-1258 (2017). DOI: 10.1216/RMJ-2017-47-4-1227

Abstract

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2=x^3+\alpha x^2-n^2 x$. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with the $\alpha =0$ case of congruent numbers. Congruent numbers are positive integers equal to the area of a right triangle with rational side lengths.

Citation

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Farzali Izadi. Foad Khoshnam. Dustin Moody. "Heron quadrilaterals via elliptic curves." Rocky Mountain J. Math. 47 (4) 1227 - 1258, 2017. https://doi.org/10.1216/RMJ-2017-47-4-1227

Information

Published: 2017
First available in Project Euclid: 6 August 2017

zbMATH: 06790012
MathSciNet: MR3689952
Digital Object Identifier: 10.1216/RMJ-2017-47-4-1227

Subjects:
Primary: 14H52
Secondary: 11G05 , 14G05 , 51M04

Keywords: congruent numbers , cyclic quadrilateral , Elliptic curves , Heron quadrilateral

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 4 • 2017
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