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Most odd-degree binary forms fail to primitively represent a square
Published online by Cambridge University Press: 08 February 2024
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.
- Type
- Research Article
- Information
- Copyright
- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
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