Quantum Statistical Mechanics of the Absolute Galois Group

Abstract

We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's dessins d'enfant, the embedding in the Grothendieck-Teichmüller group, and the Drinfeld-Ihara involution.

Three arithmetic sites

“…The degree function d of a dynamical Belyi polynomial gives a morphism of monoids d : B ✲ N × + and d : B Q ✲ N × + , and therefore geometric morphisms to the arithmetic site B ✲ A and B Q ✲ A Remains to link the Belyi-site(s) to the Conway-site. The approach below is inspired by [12,Prop. 2.25].…”