Treffer: Bost–Connes–Marcolli System for the Siegel Modular Variety

Bost–Connes–Marcolli systems are an important type of quantum statistical mechanical systems which provide connections between the ergodic theory of dynamical systems, class field theory and von Neumann algebras. In this thesis, we generalize the GL₂-system to the case of the Siegel modular variety of degree two and study its various properties. We show that this dynamical system undergoes a spontaneous symmetry breaking phase transition and classify its equilibrium extremal states at different inverse temperatures. We next study its symmetry group and derive an intertwining equality between the action by symmetries and a subgroup of the Galois group of the Siegel modular field. Finally, we study the von Neumann algebras associated to the equilibrium states and classify their types.