Thue equations that simultaneously fail the Hasse principle

Technical Summary: Thue Equations That Simultaneously Fail the Hasse Principle

Problem Statement
The paper addresses the integral Hasse principle for Thue equations of the form $F(x, y) = h$, where $F$ is an irreducible binary form with integer coefficients of degree $n \geq 3$, and $h$ is a positive integer. While Thue's theorem guarantees finitely many integer solutions for $|F(x, y)| \leq m$, the integral Hasse principle posits that if an equation has solutions in the $p$-adic integers $\mathbb{Z}_p$ for all primes $p$ (local solubility), it should have a solution in the integers $\mathbb{Z}$ (global solubility).

Previous work by Akhtari and Bhargava demonstrated that a positive proportion of Thue equations $F(x, y) = m$ fail this principle. The specific problem addressed in this paper is to refine this construction to show that, for any fixed positive integer $m$, a positive proportion of binary forms $F$ fail the integral Hasse principle simultaneously for all equations $F(x, y) = h$ where $1 \leq h \leq m$.

Methodology
The author employs a constructive counting argument based on the height of binary forms, ordered by the maximum absolute value of their coefficients. The strategy relies on three main components:

1. Upper Bounds on Solutions: The construction utilizes a result by Györy (Theorem 2.1), which provides an upper bound on the number of primitive integer solutions to Thue inequalities $|F(x, y)| \leq M$ when the discriminant $|D(F)|$ is sufficiently large relative to $M$. Specifically, if $|D(F)| \geq (3nM)^{6(n-1)}$, the number of primitive solutions is at most $32n + 11$.

2. Local-Global Construction:

   • Local Solubility: The author constructs forms that represent every integer $h \leq m$ locally everywhere (over $\mathbb{Z}_p$ for all $p$). This is achieved by ensuring the forms have simple roots modulo small primes $p_i$ (allowing lifting via Hensel's Lemma) and satisfy specific irreducibility conditions modulo large primes (ensuring smoothness of the associated projective curves and applying the Hasse-Weil bound).
   • Global Non-Solubility: The construction generates a large family of forms $\{G_j\}$ from a single "seed" form $F$. Each $G_j$ is derived by applying a sequence of linear transformations involving primes $p_1, \dots, p_k$ (where $p_i < M$). A solution to $G_j(x, y) = h$ implies a solution to a scaled equation involving $F$.

3. Counting and Density:

   • The author defines a set $\tilde{S}$ of "seed" forms satisfying specific Galois and congruence conditions (e.g., Galois group $S_n$, simple roots modulo small primes).
   • From each $F \in \tilde{S}$, the author constructs $2^k$ distinct forms $G_j$ (where $k$ is the number of primes less than $M$).
   • The paper proves that these $G_j$ forms are pairwise non-equivalent and maximal.
   • By comparing the number of forms in the constructed set $S$ (those failing the Hasse principle) against the total number of forms of a given height, the author derives a lower bound for the proportion of such forms.

Key Contributions and Results
The primary result is Theorem 1.1, which states:

Let $m$ be a positive integer. When integral binary forms of fixed degree $n \geq 3$ are ordered by height, a positive proportion of primitive maximal forms represent every positive integer less than $m$ locally everywhere but not globally.

Key technical steps supporting this result include:

• Simultaneous Failure: Unlike previous results focusing on a single $h$, this construction ensures the failure of the Hasse principle for the entire range $1 \leq h \leq m$ for the same set of forms.
• Explicit Proportion: The paper provides an explicit lower bound for the proportion of such forms. The proportion depends on $m$ and the degree $n$, derived from the ratio of the number of constructed forms to the total number of forms of a given height. The bound involves terms related to the number of primes $k$ and the discriminant growth.
• Maximality: The constructed forms are shown to be maximal (not proper subforms of other forms), which is necessary to ensure the counting argument regarding solution numbers is valid.
• Generalization: The paper notes that the construction extends to cubic forms ordered by discriminant and quartic forms ordered by their invariants $I$ and $J$.

Significance and Claims
The paper claims to refine the construction in [2] (Akhtari and Bhargava) to achieve simultaneous failure of the Hasse principle for a range of integers. The significance lies in demonstrating that the failure of the integral Hasse principle is not an isolated phenomenon for specific values of $h$ but is a robust property shared by a positive proportion of forms across a spectrum of target integers.

The author explicitly states that the work is a refinement of previous constructions and does not claim to solve the general problem of characterizing all such forms, but rather establishes the existence of a positive proportion with these specific simultaneous failure properties. The paper relies on established sieve methods (Ekedahl's sieve) and bounds from Diophantine geometry (Györy, Hasse-Weil) to validate the density of the constructed set.