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 $\text{KMS}<em>\beta$-states for inverse temperatures $\beta &gt;0$ and show that a spontaneous phase transition occurs at $\beta=3$. More precisely, we prove that the system does not admit a $\text{KMS}</em>\beta$ state for $\beta &lt; 3$ with $\beta \neq 1$, construct the explicit extremal Gibbs states for $\beta &gt;4$ and show that a unique $\text{KMS}_\beta$ state exists for every $\beta &gt;0$ with $3 &lt; \beta\leq 4$.