Congruence properties of induced representations and their applications

Abstract

Congruence properties of the representations $U_\alpha :=U^{\mathrm {PSL} (2,\mathbb {Z})}_{\chi _\alpha }$ are studied for the projective modular group $\mathrm {PSL} (2,\mathbb {Z})$ induced by a family $\chi _\alpha$ of characters for the Hecke congruence subgroup $\Gamma _0(4)$, basically introduced by A. Selberg. The interest in the representations $U_\alpha$ stems from their presence in the transfer operator approach to Selberg’s zeta function for this Fuchsian group and the 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 the $U_\alpha$ are easily shown to be equivalent to those 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 character groups of the same kind for $\Gamma (4)$. Furthermore, unlike 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 $\ker U_\alpha$, for the reason that they can have arbitrary genus $g\geq 0$, unlike the groups $\ker \chi _\alpha$, which all have genus $g=0$.