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Most odd-degree binary forms fail to primitively represent a square

Part of: Diophantine equations Curves Linear algebraic groups and related topics Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  08 February 2024

Ashvin A. Swaminathan Open the ORCID record for Ashvin A. Swaminathan [Opens in a new window]
Ashvin A. Swaminathan*
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA ashvins@alumni.princeton.edu
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Abstract

Let $F$ be a separable integral binary form of odd degree $N \geq 5$. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-$N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family $\mathscr {F}_N(f_0)$, more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$, respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

Keywords

superelliptic equationstacky curveprimitive integer solutionarithmetic statistics

MSC classification

Primary: 11D45: Counting solutions of Diophantine equations 14G05: Rational points
Secondary: 20G25: Linear algebraic groups over local fields and their integers 14H25: Arithmetic ground fields

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 3 , March 2024 , pp. 481 - 517
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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