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

Abstract

We show that the commutator equation over SL 2 (Z) satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for SL 2 (Z[ 1 p ]). The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces. Contents 1. Introduction 2. Local to global principle for the commutator word in SL 2 (Z) 2.1. Proof of Proposition 2.5 2.2. The ring Z[G m,n /G ′ m,n ] and the module G ′ m,n /G ′′ m,n . 2.3. Trace computations 2.4. Proof of Proposition 2.6 3. Lifting solutions 3.1. General D 3.1.1. An explicit analysis 3.2. The case D = Z 3.2.1. An algorithmn for (E 2 (Z)) 3.3. D = Z[ √ −d], Bianchi groups 4. Universal domains 5. Hasse failures 5.1. Hasse failures for Markoff surfaces 5.1.1. Markoff surfaces and ternary quadratic forms, SL 2 (Z)-revisited 5.1.2. Some Hasse failures revisited 5.1.3. Hasse failures over S-integers 5.2. Hasse failures for commutators Appendix A. Preliminaries B. Local analysis References