Refined limit multiplicity for varying conductor

TL;DR

This paper refines the limit multiplicity problem for automorphic representations by focusing on counting those with a specific conductor, providing a more detailed understanding of their distribution across various groups and level subgroups.

Contribution

It introduces a refined counting method for automorphic representations with fixed conductor, extending previous limit multiplicity results to a more precise setting.

Findings

Automorphic representations with fixed conductor are shown to have multiplicity one in the limit

The refined analysis applies to general classes of groups and level subgroups

Enhances understanding of the distribution of automorphic representations in the limit

Abstract

Recent results by Abert, Bergeron, Biringer et al., Finis, Lapid and Mueller, and Shin and Templier have extended the limit multiplicity property to quite general classes of groups and sequences of level subgroups. Automorphic representations in the limit multiplicity problem are traditionally counted with multiplicity according to the number of fixed vectors of a level subgroup; our goal is to perform a slightly more refined analysis and count only automorphic representations with a given conductor with multiplicity 1.