Twist-minimal trace formulas and the Selberg eigenvalue conjecture

TL;DR

This paper develops an explicit Selberg trace formula for twist-minimal Maass forms and uses it to classify certain Artin representations and verify the Selberg eigenvalue conjecture for small levels.

Contribution

It provides a fully explicit trace formula for twist-minimal Maass forms and applies it to classify Artin representations and verify the Selberg eigenvalue conjecture.

Findings

Classified small conductor even 2D Artin representations under Artin's conjecture.

Identified the smallest conductor icosahedral Artin representation as found by Doud and Moore.

Verified the Selberg eigenvalue conjecture for small level groups, improving previous results.

Abstract

We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.