Conductor zeta function for the GL(2) universal family

TL;DR

This paper establishes a Weyl law with power savings for universal families of cuspidal automorphic representations of GL(2) over Q, and for Hecke characters over any number field, using properties of the conductor zeta function.

Contribution

It introduces a method to analyze the distribution of automorphic representations and Hecke characters via the analytic properties of the conductor zeta function.

Findings

Weyl law with power savings for GL(2) automorphic families

Extension of results to Hecke characters over any number field

Method based on analytic properties of the conductor zeta function

Abstract

We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.