Bost-Connes-Marcolli system for the Siegel modular variety

Abstract

We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli $GL_2$ system. In particular we introduce the Connes-Marcolli system associated to the Siegel modular variety of degree $2$. We classify its $\text{KMS}_\beta$-states for inverse temperatures $\beta >0$ and show that a spontaneous phase transition occurs at $\beta=3$. More precisely, we prove that the system does not admit a $\text{KMS}_\beta$ state for $\beta < 3$ with $\beta \neq 1$, construct the explicit extremal Gibbs states for $\beta >4$ and show that a unique $\text{KMS}_\beta$ state exists for every $\beta >0$ with $3 < \beta\leq 4$.