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

Ashvin Swaminathan

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-$N$ \emph{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.

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

Abstract

Let be a separable integral binary form of odd degree . A result of Darmon and Granville known as ``Faltings plus epsilon'' implies that the degree- \emph{superelliptic equation} has finitely many primitive integer solutions. In this paper, we consider the family of degree- superelliptic equations with fixed leading coefficient , ordered by height. For every sufficiently large , we prove that among equations in the family , more than are insoluble, and more than 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 and , respectively, when 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 have no rational points.

Paper Structure

This paper contains 24 sections, 36 theorems, 131 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Superelliptic Stacky Curves
  3. Main Results
  4. Method of Proof
  5. Algebraic Preliminaries
  6. A Representation of the Split Odd Special Orthogonal Group
  7. Rings Associated to Binary Forms
  8. Parametrization of Square Roots of the Class of $J_F$
  9. Orbits over Fields
  10. Construction of an Integral Orbit from a Primitive Integer Solution
  11. The Construction
  12. Distinguished Points
  13. Proof of Theorem \ref{['thm-main']}
  14. Outline of the Proof
  15. Step (A): Counting Irreducible Integral Orbits of Bounded Height
  16. ...and 9 more sections

Key Result

Theorem 1

Let $f_0$ be a non-square integer. For every sufficiently large $n$, most superelliptic stacky curves in the family $\mathscr{F}_{2n+1}(f_0)$ have no primitive integer solutions. More precisely:

Figures (1)

  • Figure 1: The pencil of conics spanned by $Q_A$ and $Q_B$ in $\mathbb P_{k_{\operatorname{sep}}}^2$, with singular fibers $Q_1$, $Q_2$, and $Q_3$ having cone points $q_1$, $q_2$, and $q_3$, respectively. In this case, $\mathscr{F}_{(A,B)}(k_{\operatorname{sep}}) = Q_A \cap Q_B = \{p_1, p_2, p_3, p_4\}$. The dashed arrows display the action of the degree-$0$ divisor $(\tfrac{1}{2}\theta_1 - \tfrac{1}{2}\theta_2)$ on $\mathscr{F}_{(A,B)}$.

Theorems & Definitions (71)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Swpreprint
  • Proposition 4
  • Remark 5
  • Proposition 6: Swpreprint
  • Proposition 7
  • proof
  • Remark 8
  • Lemma 9
  • ...and 61 more