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

Published 21 Oct 2021 in math.NT and math.GR | (2110.11030v2)

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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