Bost-Connes-Marcolli system for the Siegel modular variety

2022/11/14 by Ismail Abouamal, Abouamal, Ismail
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2211.07778

openalex publication_date 2022/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli GL₂ system. In particular we introduce the Connes-Marcolli system associated to the Siegel modular variety of degree 2. We classify its KMS_β-states for inverse temperatures β>0 and show that a spontaneous phase transition occurs at β=3. More precisely, we prove that the system does not admit a KMS_β state for β< 3 with β≠ 1, construct the explicit extremal Gibbs states for β>4 and show that a unique KMS_β state exists for every β>0 with 3 < β≤ 4.

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