An Arithmetic Invariant of the Jacquet-Langlands correspondence

TL;DR

This paper investigates how the Jacquet-Langlands correspondence preserves certain measures and densities, establishing local-global compatibility and extending the notion of module densities over discrete groups.

Contribution

It introduces a new perspective on the Jacquet-Langlands correspondence by analyzing the preservation of densities and measures, connecting local and global aspects.

Findings

Global Jacquet-Langlands correspondence preserves densities over principal arithmetic groups.

Describes local-global compatibility of Plancherel and Tamagawa measures.

Generalizes dimensions over discrete groups to densities of modules.

Abstract

We describe the local-global compatibility of local Plancherel measures and the Tamagawa measure under the Jacquet-Langlands correspondence. We apply the notion of densities of modules over a discrete group, which generalizes the dimensions over a discrete group. We prove that the global Jacquet-Langlands correspondence preserves the densities over principal arithmetic groups.