Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles

TL;DR

This paper constructs infinite families of elliptic curves with specific ranks from Heron triangles, advancing understanding of their algebraic properties under certain conjectural assumptions.

Contribution

It introduces new methods to generate infinite families of elliptic curves with controlled 2-Selmer ranks from Heron triangles.

Findings

Constructed infinite families of rank 1 elliptic curves from Heron triangles.

Produced families of elliptic curves with 2-Selmer rank between 1 and 3.

Assumed finiteness of the Shafarevich-Tate group for constructions.

Abstract

Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.