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Abstract: We show that noncongruence subgroups of SL 2Z projectively equivalent tocongruence subgroups are ubiquitous. More precisely, they always exist if thecongruence subgroup in question is a principal congruence subgroup GammaN oflevel N>2, and they exist in many cases also for Gamma 0N.The motivation for asking this question is related to modular forms:projectively equivalent groups have the same spaces of cusp forms for all evenweights whereas the spaces of cusp forms of odd weights are distinct ingeneral. We make some initial observations on this phenomenon for weight 3 viageometric considerations of the attached elliptic modular surfaces.We also develop algorithms that construct all subgroups projectivelyequivalent to a given congruence subgroup and decides which of them arecongruence. A crucial tool in this is the generalized level concept ofWohlfahrt.

Autor: Ian Kiming, Matthias Schuett, Helena Verrill


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