Euclidean Geometry and Elliptic Curves

TL;DR

This paper explores the connections between specific geometric figures and families of elliptic curves with positive ranks, providing methods to construct such curves based on properties of Heron triangles, Brahmagupta quadrilaterals, and Bicentric quadrilaterals.

Contribution

It introduces novel constructions of elliptic curves with various positive ranks and torsion subgroups derived from classical geometric figures.

Findings

Constructed families of elliptic curves with positive ranks

Linked geometric properties to elliptic curve parameters

Demonstrated methods for generating elliptic curves from geometric figures

Abstract

In this paper, we demonstrate the intimate relationships among some geometric figures and the families of elliptic curves with positive ranks. These geometric figures include \textit{\textbf{Heron triangles}}, \textit{\textbf{Brahmagupta quadrilaterals}} and \textit{\textbf{Bicentric quadrilaterals}}. Firstly, we investigate the important properties of these figures and then utilizing these properties, we show that how to construct various families of elliptic curves with different positive ranks having different torsion subgroups.