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.

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