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

2023/06/27 by Yasuhiro Ishitsuka, Ishitsuka, Yasuhiro, Yoshinori Kanamura +1
Mathematics · Social Sciences · #11N45 (Secondary) #14G05 (Primary) 11G05 #14G12 #14H25 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2306.15233

openalex publication_date 2023/06/27 · openalex created_date 2023/06/29 · openalex updated_date 2026/07/28

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 ℚ_v for every place v of ℚ (resp. over 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 y²=x³-n²x by Heath-Brown, Xiong--Zaharescu and Smith.

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