Lifts of projective congruence groups, II

TY - JOUR

T1 - Lifts of projective congruence groups, II

AU - Kiming, Ian

PY - 2014

Y1 - 2014

N2 - We continue and complete our previous paper “Lifts of projective congruence groups” concerning the question of whether there exist noncongruence subgroups of SL2(ℤ) that are projectively equivalent to one of the groups Γ0(N) or Γ1(N). A complete answer to this question is obtained: In case of Γ0(N) such noncongruence subgroups exist precisely if N ∉ {3, 4, 8} and we additionally have either that 4 | N or that N is divisible by an odd prime congruent to 3 modulo 4. In case of Γ1(N) these noncongruence subgroups exist precisely if N >4. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of SL2(ℤ) that can serve as the basis for experimentation with modular forms on noncongruence subgroups.

AB - We continue and complete our previous paper “Lifts of projective congruence groups” concerning the question of whether there exist noncongruence subgroups of SL2(ℤ) that are projectively equivalent to one of the groups Γ0(N) or Γ1(N). A complete answer to this question is obtained: In case of Γ0(N) such noncongruence subgroups exist precisely if N ∉ {3, 4, 8} and we additionally have either that 4 | N or that N is divisible by an odd prime congruent to 3 modulo 4. In case of Γ1(N) these noncongruence subgroups exist precisely if N >4. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of SL2(ℤ) that can serve as the basis for experimentation with modular forms on noncongruence subgroups.

U2 - 10.1090/S0002-9939-2014-12127-7

DO - 10.1090/S0002-9939-2014-12127-7

M3 - Journal article

SN - 0002-9939

VL - 142

SP - 3761

EP - 3770

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 11

ER -