Sharp bounds for multiplicities of Bianchi modular forms

TL;DR

This paper establishes sharp bounds on the growth of Bianchi modular forms over certain number fields by connecting automorphic forms to algebraic microlocalisation and Iwasawa modules, advancing understanding of their dimensions.

Contribution

It introduces a novel approach using microlocalisation of completed homology to derive bounds on automorphic form dimensions over complex quadratic fields.

Findings

Proves a degree-one bound for the dimension of cohomological automorphic forms.

Establishes a sharp growth bound for cuspidal Bianchi modular forms.

Demonstrates finitely generated Iwasawa modules are generic under microlocalisation.

Abstract

We prove a degree-one saving bound for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on $\mathrm{SL}_2$ over any number field that is not totally real. In particular, we establish a sharp bound on the growth of cuspidal Bianchi modular forms. We transfer our problem into a question over the completed universal enveloping algebras by applying an algebraic microlocalisation of Ardakov and Wadsley to the completed homology. We prove finitely generated Iwasawa modules under the microlocalisation are generic, solving the representation theoretic question by estimating growth of Poincar\'e-Birkhoff-Witt filtrations on such modules.