Characterization of global fields by Dirichlet L-series

Abstract

We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.Date: June 15, 2017 (version 1.0).

Reconstructing the Bost–Connes semigroup actions from K-theory

“…Second, Theorem 1.1 does not mean that the isomorphism is reconstructed from K-theoretic data. Indeed, if we have an automorphism ϕ on A K as a C*-algebra over 2 P K (e.g., the action of G ab K induced from its action by multiplication on the second factor of Y K ), then we can apply the commutativity of the diagram (5) in the proof of Theorem 1.1 above for the family {ϕ F * : K * (B F K ) → K * (B F K )} to see that ϕ F * are identity maps.…”

“…For example, an isomorphism of Bost-Connes systems immediately induces a bijection between abelianized Galois groups. This problem was first considered in [7] and recently a remarkable partial solution is given by Cornelissen, de Smit, Li, Marcolli and Smit in [5,6], where it is proved that the Bost-Connes semigroup actions Y K I K and Y L I L are conjugate if and only if K is isomorphic to L.…”

“…(4) The Bost-Connes C*-algebras A K and A L are isomorphic. (5) There is a bijection P K ∼ = P L (which identifies 2 P K ∼ = 2 P L ) and an ordered KK(2 P K )-equivalence between A K and A L . (6) There is a bijection P K ∼ = P L and a family of ordered isomorphisms ϕ F : K * (B F K ) → K * (B F L ) such that ϕ F ∪{p} •∂ F,p K = ∂ F,p L •ϕ F for any finite subset F ⊂ P K ∼ = P L .…”

“…Here we say that an isomorphism of K 0 -groups is ordered if it gives a bijection of the positive cones (see for example [2,Section 6]). The precise meaning of (5) is that there is an invertible element in KK(2 P K ; A K , A L ) ( [11,Definition 4.1], see also [14,Definition 3.1]) which induces a family of ordered isomorphisms between filtered K 0 -groups [15,Definition 2.4]. The C*-algebra B F K and the homomorphism ∂ F,p K are defined in Definition 2.11 and Definition 2.14 respectively.…”

Reconstructing global fields from dynamics in the abelianized Galois group

“…Our main result says that natural equivalences of such dynamical systems for different fields (orbit equivalence, topological conjugacy) have a purely number theoretical meaning, called a "reciprocity isomorphism". Referring to the companion paper [8], this is equivalent to equality of Dirichlet L-series, and isomorphism of fields. The main result is Theorem 3.1 below.…”