Inertial multiplicity bounds for two dimensional projective representations and bounds for the number of ${\mathrm{PSL}}_2({\mathbb{F}}_q)$ and ${\mathrm{PGL}}_2({\mathbb{F}}_q)$ number fields

Abstract

We prove upper bounds for certain number field counting functions using Serre's modularity conjecture (now a theorem of Khare--Wintenberger). These results are comparable to sharp upper bounds for the number of abelian extensions with fixed or bounded discriminant, with Serre's modularity conjecture playing the role of class field theory.