Mordell-Weil ranks of families of elliptic curves parametrized by binary quadratic forms

TL;DR

This paper investigates the Mordell-Weil ranks of specific families of elliptic curves parametrized by binary quadratic forms, providing explicit bounds over number fields and detailed group structure descriptions over function fields.

Contribution

It offers new explicit lower bounds for Mordell-Weil ranks and detailed structural insights for elliptic curves parametrized by binary quadratic forms.

Findings

Explicit lower bounds for Mordell-Weil ranks over number fields

Detailed Mordell-Weil group structure over function fields

Results applicable to families of elliptic curves defined by quadratic forms

Abstract

We prove results on the Mordell--Weil rank of elliptic curves $y^2=x(x-\alpha a^2)(x-\beta b^2)$ parametrized by binary quadratic forms $\alpha a^2+\beta b^2=\gamma c^2$. We express our explicit lower bounds over number fields and offer a detailed description of the corresponding Mordell-Weil group structure in the function field case.