# MathSciDoc: An Archive for Mathematician ∫

#### Number Theorymathscidoc:1802.24001

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.
title={Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$},
Yongquan Hu. Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180208110840331107914.