A positive proportion of Thue equations fail the integral Hasse principle

TL;DR

This paper demonstrates that a positive proportion of integral binary cubic and higher degree Thue equations, ordered by discriminant or coefficients, fail the integral Hasse principle despite having local solutions everywhere.

Contribution

It establishes that a positive proportion of Thue equations of degree three and higher fail the integral Hasse principle when ordered by natural invariants.

Findings

A positive proportion of cubic Thue equations fail the integral Hasse principle.

The result extends to Thue equations of any fixed degree n ≥ 3.

Ordering by discriminant or coefficients reveals significant failures of the Hasse principle.

Abstract

For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.