The density hypothesis for horizontal families of lattices

TL;DR

This paper proves the density hypothesis for broad classes of arithmetic orbifolds from division quaternion algebras over number fields, providing uniform bounds on non-tempered representation multiplicities.

Contribution

It establishes the density hypothesis for wide families of arithmetic orbifolds with uniform bounds, advancing understanding of spectral multiplicities in this context.

Findings

Proves the density hypothesis for arithmetic orbifolds from division quaternion algebras.

Provides uniform power-saving bounds on multiplicities of non-tempered representations.

Applies to all number fields of bounded degree.

Abstract

We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects.