Twist-minimal trace formulas and the Selberg eigenvalue conjecture

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.

arXiv comment64 pages, to appear in JLMS

References in corpus (4)

• Explicit calculations with Eisenstein series
• On the order of vanishing of newforms at cusps
• Density Theorems for Exceptional Eigenvalues for Congruence Subgroups
• Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

Cited by in corpus (5)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Computing classical modular forms
• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
• Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect
• Correction to: Equidistribution of Eisenstein Series in the Level Aspect