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

TL;DR

This work investigates local-global principles for semi-integral points on Campana orbifold pairs attached to Markoff surfaces, using an adelic framework and Brauer--Manin obstructions to compare semi-integral, rational, and integral points. The authors establish that Markoff orbifold pairs satisfy the semi-integral Hasse principle and analyze weak and strong approximation off finite sets, with detailed dependence on the number and finiteness of boundary weights and on the square-ness of $m-4$. They show that algebraic Brauer--Manin obstructions often vanish for strict semi-integral points, yet failures of strong and weak approximation occur in various scenarios, and they provide explicit families exhibiting strict semi-integral points even when the affine Markoff surface has no integral points. Moreover, they obtain a counting result for the occurrence of strict semi-integral Hasse phenomena in families, using binary quadratic forms and genus theory to construct concrete examples and derive asymptotics. These results shed light on how semi-integral local-global principles behave in families of affine log Calabi–Yau varieties and give explicit instances where semi-integral points persist despite the absence of integral points.

Abstract

We study local-global principles for semi-integral points on orbifold pairs of Markoff type. In particular, we analyse when these orbifold pairs satisfy weak weak approximation, weak approximation and strong approximation off a finite set of places. We show that Markoff orbifold pairs satisfy the semi-integral Hasse principle and we measure how often such orbifold pairs have strict semi-integral points but the corresponding Markoff surface lacks integral points.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Remark 2.2
• Definition 2.3