Abstract
We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry.
Highlights
We look at the main constructions of class field theory from the point of view of topological dynamical systems
We describe the link between the noncommutative point of view and our work by considering algebraic crossed product algebras in Reconstructing global fields from dynamics in the
Is there a “Hom-form” of our main theorem, in which field homomorphisms correspond in a precise way to topological conjugacies of the dynamical systems? Secondly, in the style of Mochizuki’s absolute version of anabelian geometry, one may ask how to reconstruct a number field from its associated dynamical system, rather than the relative result about reconstructing an isomorphism of fields from an isomorphism of their dynamical systems
Summary
We look at the main constructions of class field theory from the point of view of topological dynamical systems. Neither the Dedekind zeta function (i.e., the Dirichlet L-series for the trivial character), nor the adele ring, nor the abelianized Galois groups of a global field determine that field uniquely up to isomorphism:. The dynamical system IK XK that we consider (which, after all, is a topological space with a monoid action) can be considered as some kind of substitute for the absolute Galois group. Anabelian geometry characterizes a number field by its absolute Galois group, an object whose “internal” understanding remains largely elusive and belongs to the Langlands programme. We end this introduction with a brief discussion of related work and open problems. Is there a “Hom-form” of our main theorem, in which field homomorphisms correspond in a precise way to topological conjugacies of the dynamical systems? Secondly, in the style of Mochizuki’s absolute version of anabelian geometry (cf. [22]), one may ask how to reconstruct a number field from its associated dynamical system (or L-series), rather than the relative result about reconstructing an isomorphism of fields from an isomorphism of their dynamical systems (or L-series)
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