Multiplicities of cohomological automorphic forms on GL2 and mod p representations of GL2(Qp).

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on GL2 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 GL2(Q9) as started by Barthel-Livné and Breuil, and developed by Pašk¯unas. [ABSTRACT FROM AUTHOR]