Bost-Connes-Marcolli system for the Siegel modular variety

We construct a quantum satisitical mechanical system which generalizes the Connes-Marcolli $G{L}_2$ 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 $\beta \mathrm{\ne }1$, construct the explicit extremal Gibbs states for $\beta >4$ and show that a unique ${\text{KMS}}_{\beta }$ state exists for every $\beta >0$ with $3<\beta \le 4$.