Quantum Statistical Mechanics over Function Fields

It has become increasingly evident, starting from the seminal paper of Bost and Connes [3] and continuing with several more recent developments ([8], [10], [12], [13], [22], [24]), that there is a rich interplay between quantum statistical mechanics and arithmetic. In the case of number fields, the symmetries and equilibrium states of the Bost–Connes system are closely linked to the explicit class field theory of Q, and the system constructed in [12] extends this result to the case of imaginary quadratic fields, using the relation between the arithmetic of the modular field and a 2-dimensional analog of the Bost–Connes system introduced in [10]. This leads to a far reaching generalization to Shimura varieties as developed in [22]. Moreover, very recently Benoit Jacob constructed an interesting quantum statistical mechanical system that generalizes the Bost–Connes system for function fields, using sign normalized rank one Drinfeld modules. In all of these cases, one always works with the C∗-algebra formulation of quantum statistical machanics. In the case of number fields one can extract arithmetic information by considering a suitable subalgebra (or algebra of multipliers) which is defined over Q or over a finite extension thereof. In the case of positive characteristic, one needs a different approach, which implies developing a version of quantum statistical mechanics that works when the algebra of observables is an algebra over a field extension of a function field rather than being an algebra over the complex numbers.