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An Isaac Newton Institute Workshop

Trends in Noncommutative Geometry

Phase transition in the Connes-Marcolli GL2-system

19th December 2006

Author: Sergey Neshveyev (University of Oslo)

Abstract

Connes and Marcolli have recently introduced a GL2-system, a two dimensional analogue of the Bost-Connes system associated with rational numbers. They analyzed the space of KMS-states of the system for inverse temperatures outside the critical region (1,2]. We complete their analysis by showing that for each inverse temperature in the critical region there exists a unique KMS-state. This uniqueness turned out to be related to equidistribution of Hecke points on the modular curve. On the way we develop some general tools for analyzing KMS-states on crossed product of abelian algebras by Hecke algebras. (Joint work with M. Laca and N.S. Larsen.)