Most Odd-Degree Binary Forms Fail to Primitively Represent a Square

Abstract

Let $F$ be a separable integral binary form of odd degree $N\ge 5$. A result of Darmon and Granville known as ``Faltings plus epsilon'' implies that the degree-$N$ \emph{superelliptic equation} $y^2=F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family ${\mathcal{F}}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0\in \mathbb{Z}\setminus \pm {\mathbb{Z}}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family ${\mathcal{F}}_N(f_0)$, 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.

arXiv comment38 pages