Congruence properties of induced representations and their applications

Abstract

In this paper we study congruence properties of the representations $U_α:=U^{PSL(2,\mathbb{Z})}_{χ_α}$ of the projective modular group ${\rm PSL}(2,\mathbb{Z})$ induced from a family $χ_α$ of characters for the Hecke congruence subgroup $Γ_0(4)$ basically introduced by A. Selberg. Interest in the representations $U_α$ stems from their appearance in the transfer operator approach to Selberg's zeta function for this Fuchsian group and character $χ_α$. Hence the location of the nontrivial zeros of this function and therefore also the spectral properties of the corresponding automorphic Laplace-Beltrami operator $Δ_{Γ,χ_α}$ are closely related to their congruence properties. Even if as expected these properties of $U_α$ are easily shown to be equivalent to the ones well known for the characters $χ_α$, surprisingly, both the congruence and the noncongruence groups determined by their kernels are quite different: those determined by $χ_α$ are character groups of type I of the group $Γ_0(4)$, whereas those determined by $U_α$ are such character groups of $Γ(4)$. Furthermore, contrary to infinitely many of the groups $\ker χ_α$, 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 $\ker U_α$, the reason being, that they can have arbitrary genus $g\geq 0$, contrary to the groups $\ker χ_α$, which all have genus $g=0$.