Bounds for the multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$

Abstract

We prove a power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on $\mathrm{GL}_2$ over any number field that 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 noncommutative Iwasawa modules.