## 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.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | noncommutative geometry; quantum statistical mechanics; operator algebras; C* dynamical systems; C* algebras; modular curves; Bost-Connes system; GL2 system; Hilbert's twelfth problem; real quadratic extensions; class field theory; mathematical physics, functional analysis | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Physics, Mathematics and Astronomy | ||||||

Major Option: | Mathematics | ||||||

Awards: | Apostol Award for Excellence in Teaching in Mathematics, 2019, 2020, 2021. | ||||||

Thesis Availability: | Public (worldwide access) | ||||||

Research Advisor(s): | - Marcolli, Matilde
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Thesis Committee: | - Mantovan, Elena (chair)
- Rains, Eric M.
- Ramakrishnan, Dinakar
- Marcolli, Matilde
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Defense Date: | 23 May 2022 | ||||||

Funders: |
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Record Number: | CaltechTHESIS:05252022-192257156 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:05252022-192257156 | ||||||

DOI: | 10.7907/xrba-8471 | ||||||

Related URLs: |
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ORCID: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 14622 | ||||||

Collection: | CaltechTHESIS | ||||||

Deposited By: | Jane Panangaden | ||||||

Deposited On: | 27 May 2022 23:12 | ||||||

Last Modified: | 04 Aug 2022 23:26 |

## Thesis Files

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