Sharp bounds for multiplicities of Bianchi modular forms

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.