Local-global principles for semi-integral points on Markoff orbifold pairs

Technical Summary: Local-Global Principles for Semi-Integral Points on Markoff Orbifold Pairs

Problem Statement
This paper investigates local-global principles for semi-integral points on orbifold pairs associated with Markoff surfaces. While the arithmetic of rational and integral points on Markoff surfaces ($u_1^2 + u_2^2 + u_3^2 - u_1u_2u_3 = m$) has been extensively studied—particularly regarding the Hasse principle and the density of integral points—this work extends the analysis to "semi-integral" points. These points, introduced by Campana and Darmon, interpolate between rational and integral points by imposing specific divisibility conditions on the intersection multiplicities with boundary divisors. The authors specifically address the behavior of these points on orbifold pairs $(X_m, D_\omega)$, where $X_m$ is the smooth compactification of the Markoff surface and $D_\omega$ is a boundary divisor with assigned weights $\omega_i \in \mathbb{Z}_{\ge 2} \cup \{\infty\}$. The central questions concern the validity of weak approximation, strong approximation, and the Hasse principle for these semi-integral points, and how often strict semi-integral points exist in the absence of integral points.

Methodology
The authors employ a combination of arithmetic geometry, the theory of quadratic forms, and the Brauer–Manin obstruction framework adapted for semi-integral points (as defined in [MNS24]).

1. Orbifold Setup and Definitions: The study utilizes the definition of Campana and Darmon points on orbifold pairs. A point is semi-integral if its intersection multiplicity with boundary components $D_i$ satisfies specific conditions based on the weight $\omega_i$: for Campana points, the multiplicity must be in $\mathbb{Z}_{\ge \omega_i} \cup \{0, \infty\}$; for Darmon points, it must be in $\omega_i\mathbb{Z}_{\ge 0} \cup \{ \infty \}$.
2. Local-Global Analysis: The authors analyze the density of semi-integral points in the adelic space. They distinguish between "strict" points (lying in the affine part $U = X \setminus D_{red}$) and "non-strict" points (lying on the boundary). The failure of weak weak approximation (WWA) is established for pairs with multiple finite weights using geometric integrality arguments and the Lang–Weil estimates.
3. Brauer–Manin Obstruction: The paper computes the values of local invariant maps for generators of the Brauer group of the Markoff surface. By constructing explicit adelic points with specific local invariant values, the authors determine the presence or absence of Brauer–Manin obstructions to the Hasse principle and approximation properties.
4. Explicit Construction via Quadratic Forms: To address the existence of strict semi-integral points when integral points are absent, the authors reduce the problem to the representation of integers by binary quadratic forms. They utilize genus theory and class group properties of specific discriminants (e.g., 117, $p^2+4$, 28) to prove the existence of solutions to the Markoff equation under specific congruence conditions on $m$.

Key Contributions and Results

• Failure of Approximation for Multiple Weights: The paper proves that if at least two of the weights $\omega_1, \omega_2, \omega_3$ are finite, the orbifold pair $(X_m, D_\omega)$ fails weak weak approximation for semi-integral points. This is the first result of its kind for Campana orbifolds with multiple irreducible boundary components.
• Approximation with Single Finite Weight: In the case where exactly one weight is finite, the behavior depends on the arithmetic of $m-4$. If $m-4$ is a square in $\mathbb{Q}$, weak approximation holds. If not, the pair fails Campana weak approximation; if the finite weight is even, it also fails Darmon weak approximation. These failures are attributed to Brauer–Manin obstructions.
• Strong Approximation: The authors demonstrate that strong approximation off any finite set fails if at least two weights are finite. Furthermore, if $m-4$ is not a square and at least one weight is finite, strong approximation fails off any finite set $T$ contained in the set of primes where $m-4$ is a square.
• Hasse Principle for Strict Points: The study shows that the family of Markoff orbifold pairs satisfies the semi-integral Hasse principle. Unlike the integral case, where local obstructions can prevent the existence of global points (as shown in [GS22], [LM21]), the strict semi-integral Hasse principle holds because the point $(0:0:1:1)$ is always semi-integral.
• Absence of Brauer–Manin Obstruction: For strict semi-integral points, the authors prove there is no algebraic Brauer–Manin obstruction to the Hasse principle. Under a specific arithmetic condition on $m$ (related to the field $\mathbb{Q}(\sqrt{m}, \sqrt{m-4})$), there is also no transcendental Brauer–Manin obstruction.
• Existence of Strict Semi-Integral Points without Integral Points: A significant contribution is the construction of explicit families of Markoff orbifold pairs that possess strict semi-integral points but lack integral points.
  • Theorem 1.7: The authors establish a lower bound for the number of such integers $m$ with $|m| \le B$. Specifically, if one weight is finite, the number of such $m$ is $\gg B / (\log B)^{1/2}$.
  • Explicit Families: The paper provides specific constructions (Theorems 4.1, 4.2, 4.5) using binary quadratic forms to generate these examples. For instance, for $m = 6d$ with specific congruence conditions, strict semi-integral points exist while $U_m(\mathbb{Z}) = \emptyset$.

Significance and Claims
The paper claims to be the first to systematically study local-global principles for Campana orbifolds with multiple irreducible boundary components. It highlights a fundamental discrepancy between the behavior of integral points and strict semi-integral points: while integral points on Markoff surfaces frequently fail the Hasse principle (existing locally but not globally), the corresponding strict semi-integral points on the orbifold pairs satisfy the Hasse principle.

The authors position their work as a step toward understanding the failures of local-global principles in families of affine varieties. By quantifying the frequency of cases where strict semi-integral points exist despite the absence of integral points, they provide a concrete subfamily of Markoff orbifold pairs that serves as a test case for the lack of a general theory predicting Hasse failures for integral points. The results suggest that semi-integral points can "repair" the failure of the Hasse principle observed in the integral setting, offering a new perspective on the arithmetic of Markoff surfaces. The paper does not claim to provide a general theory for all affine varieties but rather offers specific, illustrative results for the Markoff case that may inform future generalizations.