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Congruence properties of induced representations and their applications

Authors: D. Mayer, A. Momeni and A. Venkov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 4.
Journal: St. Petersburg Math. J. 26 (2015), 593-606
MSC (2010): Primary 20H05
Published electronically: May 6, 2015
MathSciNet review: 3289187
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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$.

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Additional Information

D. Mayer
Affiliation: Lower Saxony Professorship, TU Clausthal, 38678 Clausthal-Zellerfeld

A. Momeni
Affiliation: Institute of Theoretical Physics, TU Clausthal, 38678 Clausthal-Zellerfeld

A. Venkov
Affiliation: Institute for Mathematics, and Centre for Quantum Geometry (QGM), University of Aarhus, 8000 Aarhus C

Keywords: Selberg's character, induced representation, congruence character, congruence representation
Received by editor(s): May 27, 2013
Published electronically: May 6, 2015
Additional Notes: The second author would like to thank the Centre for Quantum Geometry for Moduli Spaces (QGM) for the financial support, kind hospitality and providing good working conditions
This work was supported by the Danish National Research Foundation Centre of Excellence “Centre for Quantum Geometry of Moduli Spaces (QGM)” and the Volkswagenstiftung through a Lower Saxony Professorship
Article copyright: © Copyright 2015 American Mathematical Society