Q-lattices: Quantum statistical mechanics and Galois theory

We review our recent results on the noncommutative geometry of Q-lattices modulo commensurability. We discuss the cases of 1-dimensional and 2-dimensional Q-lattices. In the first case, we show that, by considering commensurability classes of 1-dimensional Q-lattices up to scaling, one recovers the Bost-Connes quantum statistical mechanical system, whose zero temperature KMS states intertwine the symmetries of the system with the Galois action of Gal(Q/Q). In the 2-dimensional case, commensurability classes of Q-lattices up to scaling give rise to another quantum statistical mechanical system, whose symmetries are the automorphisms of the modular field, and whose (generic) zero temperature KMS states intertwine the action of these symmetries with the Galois action on an embedding in C of the modular field. Following our joint work with Ramachandran, we then show how the noncommutative spaces associated to commensurability classes of Q-lattices up to scale have a natural geometric interpretation as noncommutative versions of the Shimura varieties Sh(GL_1,{±1}) in the Bost-Connes case and Sh(GL_2, H^±) in the case of the GL_2 system. We also show how this leads naturally to the construction of a system generalizing the Bost-Connes system that fully recovers the explicit class field theory of imaginary quadratic fields