Rocky Mountain Journal of Mathematics

Heron quadrilaterals via elliptic curves

Farzali Izadi, Foad Khoshnam, and Dustin Moody

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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.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1227-1258.

Dates
First available in Project Euclid: 6 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1501984947

Digital Object Identifier
doi:10.1216/RMJ-2017-47-4-1227

Mathematical Reviews number (MathSciNet)
MR3689952

Zentralblatt MATH identifier
06790012

Subjects
Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 14G05: Rational points 51M04: Elementary problems in Euclidean geometries

Keywords
Heron quadrilateral cyclic quadrilateral congruent numbers elliptic curves

Citation

Izadi, Farzali; Khoshnam, Foad; Moody, Dustin. Heron quadrilaterals via elliptic curves. Rocky Mountain J. Math. 47 (2017), no. 4, 1227--1258. doi:10.1216/RMJ-2017-47-4-1227. https://projecteuclid.org/euclid.rmjm/1501984947


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