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

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.

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References in corpus (3)

• Integral points on Markoff type cubic surfaces
• Integral Hasse principle and strong approximation for Markoff surfaces
• Brauer-Manin obstruction for Markoff surfaces

Cited by in corpus (2)

• Integral points on Markoff type cubic surfaces
• Tautological characteristic classes II: the Witt class