Quantum Statistical Mechanics of Shimura Varieties

Abstract

The Bost-Connes and Connes-Marcolli C*-dynamical systems are seen to be associated to the Shimura varieties for GL(1) and GL(2), respectively. In this thesis we carry out the construction of Bost-Connes-Marcolli systems (consisting of a groupoid and an associated C*-dynamical system) for general Shimura varieties. We study the detailed structure of the underlying groupoid, attach to it various zeta functions (that coincide with statistical-mechanical partition functions and, in certain cases, classical zeta functions), and analyze its low-temperature KMS states. We also study various special cases. Our Shimura-variety approach provides a unified treatment of such C*-dynamical systems, and for the first time allows for the construction of a Bost-Connes system for a general number field F that admits symmetry by the group of connected components of the idele class group of F, and recovers the Dedekind zeta function as a partition function. One noteworthy (and rather crucial) ingredient in our constructions is a reductive monoid for the reductive group associated to the Shimura variety. Such monoids, which have been studied by Lenner, Putcha, Vinberg, and Drinfeld, are closely related to reductive groups, but (to the best of our knowledge) have hitherto played little role in the theory of Shimura varieties. Our work reveals their relation to noncommutative spaces.