An Arithmetic Invariant of the Jacquet-Langlands correspondence

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.

This version has been shortened; the representation-theoretic and operator-algebraic content has been moved to arXiv:2503.13929 (published in Communications in Mathematical Physics, Volume 407, Article 100, 2026)