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HERON QUADRILATERALS VIA ELLIPTIC CURVES.

Farzali Izadi1, Foad Khoshnam2, Dustin Moody3

  • 1Department of Mathematics, Faculty of Science, Urmia University, P. O. Box 165, Urmia 5715799313, Iran.

Rocky Mountain Journal of Mathematics
|July 6, 2019
PubMed
Summary

This study connects Heron quadrilaterals, cyclic quadrilaterals with rational properties, to a specific family of elliptic curves. This research generalizes prior work on Heron triangles and explores elliptic curve properties and congruent numbers.

Keywords:
11G0514G052010 AMS Mathematics subject classificationHeron quadrilateralPrimary 14H52Secondary 51M04congruent numberscyclic quadrilateralelliptic curves

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Diophantine Geometry

Background:

  • Heron quadrilaterals are cyclic quadrilaterals with rational side lengths and area.
  • Previous work established connections between Heron triangles and elliptic curves.
  • Congruent numbers are integers that are areas of right triangles with rational sides.

Purpose of the Study:

  • To establish a correspondence between Heron quadrilaterals and a family of elliptic curves.
  • To generalize existing connections between Heron triangles and elliptic curves.
  • To study the properties of these elliptic curves, including their torsion groups and ranks, and their relation to congruent numbers.

Main Methods:

  • Establishing a novel correspondence between Heron quadrilaterals and elliptic curves of the form y^2 = x^3 + αx^2 - n^2x.
  • Analyzing the torsion groups and ranks of the studied elliptic curves.
  • Investigating the specific case where α = 0 to explore the connection with congruent numbers.

Main Results:

  • A direct correspondence is established between Heron quadrilaterals and the specified family of elliptic curves.
  • The study provides a generalized framework building upon prior work on Heron triangles.
  • The analysis reveals insights into the structure of these elliptic curves and their relationship to number theory concepts like congruent numbers.

Conclusions:

  • The established correspondence provides a new lens for studying Heron quadrilaterals through the theory of elliptic curves.
  • The research deepens the understanding of the interplay between geometry, number theory, and algebraic geometry.
  • Further investigation into the torsion and rank properties of these curves can yield significant results in number theory.