On the proportions of soluble forms in some families of locally soluble binary quartic forms

TL;DR

This paper investigates the proportion of soluble integral binary quartic forms within locally soluble forms, providing estimates for specific subfamilies and building on prior results related to elliptic curves.

Contribution

It offers new estimates for the proportions of soluble forms in certain subfamilies, extending previous work by Bhargava and Bhargava--Ho.

Findings

Proportion estimates for specific subfamilies of binary quartic forms

Extension of prior results on soluble forms

Utilizes results from elliptic curve research

Abstract

An integral binary quartic form is said to be locally soluble (resp. soluble) if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$ for every place $v$ of $\mathbb{Q}$ (resp. over $\mathbb{Q}$). We consider the proportion of soluble integral binary quartic forms in locally soluble forms. Bhargava showed the proportion is positive when one considers all binary quartics, and Bhargava--Ho proved the proportion is zero for a subfamily. In this paper, we estimate the proportions for some other subfamilies. It relies on results for elliptic curves $y^2=x^3-n^2x$ by Heath-Brown, Xiong--Zaharescu and Smith.