A Caltech Library Service

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


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.


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.))
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
Thesis Committee:
  • Mantovan, Elena (chair)
  • Rains, Eric M.
  • Ramakrishnan, Dinakar
  • Marcolli, Matilde
Defense Date:23 May 2022
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)PGSD3-490063-2016
Record Number:CaltechTHESIS:05252022-192257156
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for chapter 3
Panangaden, Jane Mariam0000-0002-6214-2130
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14622
Deposited By: Jane Panangaden
Deposited On:27 May 2022 23:12
Last Modified:04 Aug 2022 23:26

Thesis Files

[img] PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page