Heron Quadrilaterals via elliptic curves

TL;DR

This paper establishes a novel connection between Heron quadrilaterals and a specific family of elliptic curves, extending previous work on Heron triangles, and explores their properties and relation to congruent numbers.

Contribution

It introduces a new correspondence between Heron quadrilaterals and elliptic curves, generalizing prior triangle-based results and analyzing their torsion groups, ranks, and link to congruent numbers.

Findings

Established a correspondence between Heron quadrilaterals and elliptic curves.

Analyzed torsion groups and ranks of these elliptic curves.

Connected congruent numbers to Heron quadrilaterals via elliptic curves.

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 $y^2 = x3+/alpha x^2-n^2x.$ 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 congruent numbers, which are the /alpha = 0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths. This is a new characterization of congruent numbers in terms of Heron Quadrilaterals.