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Quantum Statistical Mechanics and the Boundary of Modular Curves

Marcolli, Matilde and Panangaden, Jane (2020) Quantum Statistical Mechanics and the Boundary of Modular Curves. . (Unpublished)

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The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL₂ and an analog for SL₂. The structure of KMS states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:The first author is partially supported by NSF grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 and Accelerator Supplement grant RGPAS-2018-522593, and the Perimeter Institute for Theoretical Physics.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPIN-2018-04937
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPAS-2018-522593
Perimeter Institute for Theoretical PhysicsUNSPECIFIED
Record Number:CaltechAUTHORS:20210825-184601111
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110544
Deposited By: George Porter
Deposited On:25 Aug 2021 22:17
Last Modified:23 Sep 2021 18:38

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