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

Citation

Panangaden, Jane Mariam (2022) Quantum Statistical Mechanics, Noncommutative Geometry, and the Boundary of Modular Curves. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xrba-8471. https://resolver.caltech.edu/CaltechTHESIS:05252022-192257156

Abstract

The Bost-Connes system is a C*-dynamical system whose partition function, KMS states, and symmetries are related to the explicit class field theory of the field of rational numbers. In particular, its zero-temperature KMS states, when evaluated on certain points in an arithmetic sub-algebra, yield the generators of the maximal abelian extension of the rationals. The Bost-Connes system can be viewed in terms of a geometric picture of 1-dimensional Q-lattices. The GL₂ system is an extension of this idea to the setting of 2-dimensional Q-lattices. A specialization of the GL₂-system introduced in by Connes, Marcolli, and Ramachandran, 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 GL₂2-system. In this viewpoint we see the projective line under a certain PGL(2,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 noncommutative 2-tori. This moduli space of the non-commutative 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 GL₂ case. We show that the ground states when evaluated on points in the arithmetic algebra give pairings of the limiting modular symbols introduced by Manin and Marcolli 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, which are constructed using Kuga modular varieties, which are non-singular projective varieties over the modular curves. We subject these modular symbols to a limiting procedure. We then show, using the coding space setting of Kessenbohmer and Stratmann, that these limiting modular symbols can be written as a Birkhoff ergodic average everywhere.

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