Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

TL;DR

This paper investigates the local-global principles for commutator equations in special linear groups, revealing a reciprocity obstruction on Markoff surfaces that causes failures in certain algebraic settings.

Contribution

It establishes a profinite local-global principle for $ ext{SL}_2(bZ)$ and identifies the reciprocity obstruction causing failures over $ ext{SL}_2(bZ[rac{1}{p}])$ in relation to Markoff surfaces.

Findings

Profinite local-global principle holds for $ ext{SL}_2(bZ)$

Failure occurs over $ ext{SL}_2(bZ[rac{1}{p}])$ due to reciprocity obstruction

Failure linked to Hasse Principle violations on cubic Markoff surfaces

Abstract

We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $ \mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.