Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 96-120 |
| Seitenumfang | 25 |
| Fachzeitschrift | Journal of the London Mathematical Society |
| Jahrgang | 83 |
| Ausgabenummer | 1 |
| Publikationsstatus | Veröffentlicht - Feb. 2011 |
Abstract
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
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in: Journal of the London Mathematical Society, Jahrgang 83, Nr. 1, 02.2011, S. 96-120.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Lifts of projective congruence groups
AU - Kiming, Ian
AU - Schütt, Matthias
AU - Verrill, Helena A.
PY - 2011/2
Y1 - 2011/2
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=78751497790&partnerID=8YFLogxK
UR - https://arxiv.org/abs/0905.4798
U2 - 10.1112/jlms/jdq062
DO - 10.1112/jlms/jdq062
M3 - Article
AN - SCOPUS:78751497790
VL - 83
SP - 96
EP - 120
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
SN - 0024-6107
IS - 1
ER -