Unconditional bounds for the multiplicity of automorphic forms of cohomological type on GL_2

TL;DR

This paper establishes unconditional bounds on the growth of cohomological automorphic forms on GL_2 over non-totally real fields, using p-adic cohomology and Iwasawa theory techniques.

Contribution

It provides the first unconditional power savings for the dimension of these automorphic forms, extending previous results to more general number fields.

Findings

Proves power saving bounds for automorphic form dimensions

Uses p-adic cohomology and Iwasawa modules in the proof

Applies to non-totally real number fields

Abstract

We prove an unconditional power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL_2 over any number field which is not totally real. Our proof involves the theory of p-adically completed cohomology developed by Calegari and Emerton, and a bound for the growth of coinvariants in certain finitely generated non-commutative Iwasawa modules.