2012/10/22 by Dieter Mayer, Mayer, Dieter, Arash Momeni +3
Mathematics · #11F70 (Primary) 30F35 #20C15 #20E40 (Secondary) #20H05 #20H10 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F70 #msc:20C15 #msc:20E40 #msc:20H05 #msc:20H10 #msc:30F35
paper · pdf · doi:10.48550/arxiv.1210.5979
v2: 15 pages; introduction revised; arguments after formula 4.20 in page 12 completed; new references added
arxiv created 2013/05/22 · arxiv updated 2013/05/23
In this paper we study congruence properties of the representations Uα:=UPSL(2,ℤ)χα of the projective modular group \rm PSL(2,ℤ) induced from a family χα of characters for the Hecke congruence subgroup Γ0(4) basically introduced by A. Selberg. Interest in the representations Uα stems from their appearance in the transfer operator approach to Selberg's zeta function for this Fuchsian group and character χα. Hence the location of the nontrivial zeros of this function and therefore also the spectral properties of the corresponding automorphic Laplace-Beltrami operator ΔΓ,χα are closely related to their congruence properties. Even if as expected these properties of Uα are easily shown to be equivalent to the ones well known for the characters χα, surprisingly, both the congruence and the noncongruence groups determined by their kernels are quite different: those determined by χα are character groups of type I of the group Γ0(4), whereas those determined by Uα are such character groups of Γ(4). Furthermore, contrary to infinitely many of the groups ker χα, 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 Uα, the reason being, that they can have arbitrary genus g≥ 0, contrary to the groups ker χα, which all have genus g=0.