We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{ GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main tool of the proof is the mod $p$ representation theory of $\mathrm{ GL}_2(\mathbb{ Q}_p)$ as started by Barthel-Livne and Breuil, and developed by Paskunas.
Categories: math.NT math.RT