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

Author: Panangaden, Jane Mariam

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Marcolli, Matilde

Committee Members: Mantovan, Elena; Rains, Eric M.; Ramakrishnan, Dinakar; Marcolli, Matilde

Option: Mathematics

DOI: 10.7907/xrba-8471

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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