Ergodicity of the action of K* on A_K

TL;DR

This paper proves that the action of the multiplicative group of a global field on its adele class space is ergodic, extending known results from the rational numbers to all global fields, both number and function fields.

Contribution

It establishes the ergodicity of the action of K* on A_K for all global fields, generalizing previous results known only for Q.

Findings

Ergodicity holds for all global fields K.

The result applies to both number fields and function fields.

Extends classical understanding of adele class space actions.

Abstract

Connes gave a spectral interpretation of the critical zeros of zeta- and L-functions for a global field K using a space of square integrable functions on the space A_K/K* of adele classes. It is known that for K=Q the space A_K/K* cannot be understood classically, or in other words, the action of Q* on A_Q is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.