A positive proportion of Thue equations fail the integral 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 $n\ge 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.

Previously cited as "A positive proportion of locally soluble Thue equations are globally insoluble", Two typos are fixed and small mathematical error in Section 4 is corrected

References in corpus (4)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• The geometric sieve and the density of squarefree values of invariant polynomials
• Failures of the Integral Hasse Principle for Affine Quadric Surfaces
• On the number of integral binary $n$-ic forms having bounded Julia invariant

Cited by in corpus (2)

• A positive proportion of locally soluble quartic Thue equations are globally insoluble
• The average number of integral points on the congruent number curves