Class field theory as a dynamical system

TL;DR

This paper explores the connection between class field theory and dynamical systems, showing how global fields can be characterized by dynamical systems and relating this to geometric analogs in Riemannian geometry.

Contribution

It introduces a novel perspective by linking class field theory to dynamical systems and extends this analogy to Riemannian geometry, providing new insights into field determination and geometric metrics.

Findings

Global fields are determined by specific dynamical systems.

Abelian L-series encode information about these fields.

Analogies with Riemannian geometry lead to a new metric in manifold space.

Abstract

This is the text from a talk at the Arbeitstagung 2011, which can serve as an introduction to arxiv:1009.0736 and arXiv:1007.0907. I first discuss how a global field is determined by a certain dynamical system, and how this relates to abelian L-series determining those fields. I then discuss an analog in Riemannian geometry, and how it leads to a metric in the space of closed Riemannian manifolds.