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

TL;DR

This paper investigates local-global principles for semi-integral points on Markoff orbifold pairs, demonstrating the semi-integral Hasse principle and analyzing the distribution of semi-integral points versus integral points.

Contribution

It establishes the semi-integral Hasse principle for Markoff orbifold pairs and examines conditions under which semi-integral points exist without integral points.

Findings

Markoff orbifold pairs satisfy the semi-integral Hasse principle.

Analysis of weak approximation properties for these pairs.

Identification of cases with semi-integral points but no 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.