Bounds for multiplicities of automorphic forms of cohomological type on GL_2

TL;DR

This paper establishes bounds and restrictions on the weights and multiplicities of cohomological automorphic forms on GL_2 over number fields, with implications for hyperbolic 3-manifolds.

Contribution

It provides new bounds on automorphic form weights and multiplicities, including a conditional power saving for hyperbolic 3-manifolds.

Findings

Restrictions on weights of cuspidal cohomological automorphic forms

Conditional power saving for multiplicities in hyperbolic 3-manifolds

Results applicable to automorphic forms arising from quaternion algebras

Abstract

We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological automorphic forms and a conditional power saving for their multiplicities when the associated locally symmetric space is a hyperbolic 3-manifold.