Quantum Statistical Mechanics, L-series and Anabelian Geometry

It is known that two number fields with the same Dedekind zeta function  are not necessarily isomorphic. The zeta function of a number field can be interpreted as the  partition function of an associated quantum statistical mechanical system. I will explain two results:  (1) Isomorphism of number fields is the same as isomorphism of these associated systems.  (Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another  version of Grothendieck's "anabelian" program, much like the Neukirch-Uchida theorem characterizes isomorphism of  number fields by topological isomorphism of their associated absolute Galois groups.)  (2) If there is an isomorphism of character groups of the abelianized Galois groups of the two number fields that  induces an equality of all corresponding L-series (not just the zeta function), then the number fields are isomorphic. This follows from the first result - de Smit and Lenstra have in the meanwhile found direct number theoretical proofs  of this second result. (joint work with Matilde Marcolli, arxiv:1009.0736)