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arXiv · 2306.15233

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

Yasuhiro Ishitsuka·Yoshinori Kanamura
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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.

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Yasuhiro Ishitsuka, Yoshinori Kanamura. 2023-06-27. On the proportions of soluble forms in some families of locally soluble binary quartic forms. https://arxiv.org/abs/2306.15233

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