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Lifts of projective congruence groups

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  • University of Copenhagen
  • Louisiana State University

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Original languageEnglish
Pages (from-to)96-120
Number of pages25
JournalJournal of the London Mathematical Society
Volume83
Issue number1
Publication statusPublished - Feb 2011

Abstract

We show that noncongruence subgroups of SL2() that are projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Γ(N) of level N>2, and they also exist in many cases for Γ0(N). The motivation for asking this question is related to modular forms: projectively equivalent groups have the same spaces of cusp forms for all even weights, whereas the spaces of cusp forms of odd weights are distinct in general. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups that are projectively equivalent to a given congruence subgroup and decide which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.

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Lifts of projective congruence groups. / Kiming, Ian; Schütt, Matthias; Verrill, Helena A.
In: Journal of the London Mathematical Society, Vol. 83, No. 1, 02.2011, p. 96-120.

Research output: Contribution to journal › Article › Research › peer review

Kiming I, Schütt M, Verrill HA. Lifts of projective congruence groups. Journal of the London Mathematical Society. 2011 Feb;83(1):96-120. doi: 10.1112/jlms/jdq062
Kiming, Ian ; Schütt, Matthias ; Verrill, Helena A. / Lifts of projective congruence groups. In: Journal of the London Mathematical Society. 2011 ; Vol. 83, No. 1. pp. 96-120.
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