On Bost-Connes systems, Endomotives and Lambda-rings

The Bost-Connes system is a C*-dynamical system whose dynamics realize the class field theory of Q.
The analogous construction of a C*-dynamical system for an arbitrary number field K is known, but to
realize again the class field theory of K through the dynamics of this system requires the construction
of a distinguished arithmetic subalgebra. Until recently the construction of such arithmetic subalgebras
was only known for K an imaginary quadratic field. In our talk we will explain how to construct such
arithmetic subalgebras for arbitrary number fields. The main ingredients of our construction will be
the theory of Endomotives, introduced by Connes, Consani and Marcolli, and a classification result
of Borger and de Smit of certain Lambda-rings in terms of the Deligne-Ribet monoid.