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

TL;DR

This paper establishes a new upper bound on the dimension of cohomological automorphic forms on $ ext{GL}_2$ over certain number fields, utilizing mod $p$ representation theory of $ ext{GL}_2( ext{Q}_p)$.

Contribution

It provides an improved upper bound for automorphic form dimensions by applying advanced mod $p$ representation theory techniques.

Findings

New upper bound for automorphic form dimensions

Application of mod $p$ representation theory of $ ext{GL}_2( ext{Q}_p)$

Extension of previous bounds by Marshall

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 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.