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Congruence properties of induced representations and their applications

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  • D. Mayer, Tech Univ Clausthal, TU Clausthal
  • ,
  • A. Momeni, Tech Univ Clausthal, TU Clausthal, Inst Theoret Phys
  • ,
  • A. Venkov

Congruence properties of the representations U-alpha := U-chi alpha(PSL(2,Z)) are studied for the projective modular group PSL(2, 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 kerU(alpha), for the reason that they can have arbitrary genus g >= 0, unlike the groups ker(chi alpha), which all have genus g = 0.

Original languageEnglish
JournalSt. Petersburg Mathematical Journal
Pages (from-to)593-606
Number of pages14
Publication statusPublished - 2015

Bibliographical note

Also published in Algebra i Analiz, tom 26 (2014), nomer 4.

    Research areas

  • Selberg's character, induced representation, congruence character, congruence representation, MODULAR GROUP, SUBGROUPS, CLASSIFICATION, ATKIN

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