Bounds for the multiplicities of cohomological automorphic forms on \(\mathrm{GL}_2\) (Q431650)

Let \(F\) be a number field that is not totally real, the paper considers the bound of the dimension of the space of cusp forms on \(\mathrm{GL}_2\) with fixed level and growing weight. There is a ``trivial'' bound \(\Delta({\mathbf d})\) (determined by the weight \({\mathbf d}\)) which is given by the trace formula method. The paper breaks this trivial bound and gives a bound of the form \(\min({\mathbf d})^{-\frac13+\varepsilon}\Delta({\mathbf d})\). When \(F\) is a totally real number field, previously Shimizu established an asymptotic formula again in terms of \(\Delta({\mathbf d})\).NEWLINENEWLINEThe automorphic forms considered are tempered but not in the discrete series. Nontrivial bounds for the dimension of the space of such forms are quite rare. The proof uses the theory of \(p\)-adically completed cohomology developed by \textit{F. Calegari} and \textit{M. Emerton} [Ann. Math. (2) 170, No. 3, 1437--1446 (2009; Zbl 1195.22015)].