Heronian elliptic curves and the size of the $2$-Selmer group

TL;DR

This paper investigates the structure of the 2-Selmer group for Heronian elliptic curves linked to Heron triangles with specific area and angle conditions, extending the understanding of Selmer groups beyond congruent number cases.

Contribution

It introduces a new analysis of the 2-Selmer group for a class of Heronian elliptic curves related to particular triangle parameters, generalizing previous work on congruent number elliptic curves.

Findings

Characterization of the 2-Selmer group structure for these elliptic curves

Connection between triangle parameters and elliptic curve properties

Extension of Selmer group analysis to new classes of Heronian curves

Abstract

A generalization of the congruent number problem is to find positive integers $n$ that appear as the areas of Heron triangles. Selmer group of a congruent number elliptic curve has been studied quite extensively. Here, we look into the $2$-Selmer group structure for Heronian elliptic curves associated with Heron triangles of area $n$ and one of the angle $\theta$ such that $\tan \frac{\theta}{2} = n^{-1}$ and $n^{2}+1=2q$ for some prime $q$.