Quantum Statistical Mechanics, Noncommutative Geometry, and the Boundary of Modular Curves

Abstract

The Bost-Connes system [BC95] is a C ^∗ -dynamical system whose partition function, KMS states, and symmetries are related to the explicit class field theory of Q. In particular, its zero-temperature KMS states, when evaluated on certain points in an arithmetic sub-algebra, yield the generators of Q^cycl. The Bost-Connes system can be viewed in terms of a geometric picture of 1-dimensional Q-lattices. The GL2- system [CM04] is an extension of this idea to the setting of 2-dimensional Q-lattices. A specialization of the GL₂-system introduced in [CMR06] is related in a similar way to the explicit class field theory of imaginary quadratic extensions.

Inspired by the philosophy of Manin’s real multiplication program, we define a boundary version of the GL2-system. In this viewpoint we see P ¹ (R) under a certain PGL2(Z) action (which is related to the shift of the continued fraction expansion) as a moduli space characterizing degenerate elliptic curves. These degenerate elliptic curves can be realized as non-commutative 2-tori. This moduli space of the noncommutative tori is interpreted as an “invisible” boundary of the moduli space of elliptic curves. In fact, we define a family of such boundary GL₂ systems indexed by a choice of continued fraction algorithm. We analyze their partition functions, KMS states, and ground states. We also define an arithmetic algebra of unbounded multipliers in analogy with the GL₂case. We show that the ground states when evaluated on points in the arithmetic algebra give pairings of the limiting modular symbols of [MM02] with weight-2 cusp forms.

We also begin the project of extending this picture to the higher weight setting by defining a higher-weight limiting modular symbol. We use as a starting point the Shokurov modular symbols [Sho81a], which are constructed using Kuga varieties over the modular curves. We subject these modular symbols to a limiting procedure. We then show, using the coding space setting of [KS07b], that these limiting modular symbols can be written as a Birkhoff ergodic average everywhere.