Quantum Statistical Mechanics, L-series and Anabelian Geometry

TL;DR

This paper establishes a deep connection between number field isomorphisms and their associated quantum statistical systems, extending the anabelian perspective and linking L-series to field isomorphisms.

Contribution

It proves that isomorphism of number fields corresponds to isomorphism of their quantum statistical mechanical systems and relates L-series equality to field isomorphism.

Findings

Number fields are characterized by their associated quantum systems.

Isomorphism of systems implies isomorphism of number fields.

Equality of all L-series implies number field isomorphism.

Abstract

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.