Quantum Statistical Mechanics, L-series and Anabelian Geometry

Gunther Cornelissen, Matilde Marcolli

Published 2010-09-03, updated 2011-04-20Version 5

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, which is a C*-algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that 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. In the second part of the paper, we use these systems to prove the following. If there is an isomorphism of character groups (viz., Pontrjagin duals) 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 is also equivalent to the purely algebraic statement that there exists a topological group isomorphism as a above and a norm-preserving group isomorphism between the ideals of the fields that is compatible with the Artin maps via the other map.