Congruence properties of induced representations

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In this paper we study representations of the projective modular group induced from the Hecke congruence group of level 4 with Selberg's character. We show that the well known congruence properties of Selberg's character are equivalent to the congruence properties of the induced representations. Concerning this congruence property, it turns out that working with the induced representations is easier than with Selberg's character itself. We also show that the kernels of the induced representations determine an infinite sequence of noncongruence groups, whose noncongruence property can not be detected by Zograf's geometric method. They belong to the class of character groups of type $\rm I$ for the principal congruence subgroup $\Gamma(4)$ and have, contrary to the noncongruence groups determined by Selberg's character which all have genus $g=0$, arbitrary genus $g\geq 0$.
Original language English arXiv.org 1210.5979 14 Published - 2012

Research areas

• Number Theory (math.NT), 11F70 (Primary) 30F35, 20H05, 20C15, 20H10, 20E40 (Secondary)

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