Distribution of Mordell--Weil ranks of families of elliptic curves

TL;DR

This paper investigates the distribution of Mordell--Weil ranks in a family of elliptic curves parametrized by coprime polynomials related to the Pythagorean conic, completing previous partial results with general proofs.

Contribution

It provides the complete characterization of Mordell--Weil rank distribution for a broad family of elliptic curves, extending earlier special case analyses.

Findings

Distribution patterns of ranks established

General results proved for the family of curves

Completes previous partial analyses

Abstract

We discuss the distribution of Mordell--Weil ranks of the family of elliptic curves $y^2=(x+\alpha f^2)(x+\beta b g^2)(x+\gamma h^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and $\alpha,\beta,\gamma$ are elements from $\overline{\mathbb{Q}}$. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.