HERON QUADRILATERALS VIA ELLIPTIC CURVES

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.