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 SL₂ 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 microlocalization of Ardakov and Wadsley to the completed homology. We prove finitely generated Iwasawa modules under the microlocalization are generic, solving the representation theoretic question by estimating growth of Poincaré--Birkhoff--Witt filtrations on such modules.