Congruence properties of induced representations and their applications

TL;DR

This paper investigates the congruence properties of induced representations of the projective modular group, revealing differences between groups determined by characters and those by induced representations, with implications for spectral theory and automorphic forms.

Contribution

It establishes the distinct congruence properties of groups derived from induced representations compared to those from characters, especially regarding their genus and geometric methods.

Findings

Groups from induced representations can have arbitrary genus.

Groups from characters are always genus zero.

Different congruence properties influence spectral analysis of automorphic forms.

Abstract

In this paper we study congruence properties of the representations $U_\alpha:=U^{PSL(2,\mathbb{Z})}_{\chi_\alpha}$ of the projective modular group ${\rm PSL}(2,\mathbb{Z})$ induced from a family $\chi_\alpha$ of characters for the Hecke congruence subgroup $\Gamma_0(4)$ basically introduced by A. Selberg. Interest in the representations $U_\alpha$ stems from their appearance in the transfer operator approach to Selberg's zeta function for this Fuchsian group and character $\chi_\alpha$. Hence the location of the nontrivial zeros of this function and therefore also the spectral properties of the corresponding automorphic Laplace-Beltrami operator $\Delta_{\Gamma,\chi_\alpha}$ are closely related to their congruence properties. Even if as expected these properties of $U_\alpha$ are easily shown to be equivalent to the ones well known for the characters $\chi_\alpha$, surprisingly, both the congruence and the noncongruence groups determined by their kernels are quite different: those determined by $\chi_\alpha$ are character groups of type I of the group $\Gamma_0(4)$, whereas those determined by $U_\alpha$ are such character groups of $\Gamma(4)$. Furthermore, contrary to infinitely many of the groups $\ker \chi_\alpha $, whose noncongruence properties follow simply from Zograf's geometric method together with Selberg's lower bound for the lowest nonvanishing eigenvalue of the automorphic Laplacian, such arguments do not apply to the groups $\ker U_\alpha$, the reason being, that they can have arbitrary genus $g\geq 0$, contrary to the groups $\ker \chi_\alpha$, which all have genus $g=0$.