Multiplicities of cohomological automorphic forms on ${\mathrm{GL}}_2$ and mod $p$ representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$

Abstract

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 in [Ann. of Math. (2) 175 (2012)]. The main tool of the proof is the mod $p$ representation theory of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ as started by Barthel–Livné and Breuil, and developed by Paškūnas.