Independent generators for congruence subgroups of Hecke groups

1995 ◽  
Vol 220 (1) ◽  
pp. 569-594 ◽  
Author(s):  
Mong-Lung Lang ◽  
Chong-Hai Lim ◽  
Ser-Peow Tan
Keyword(s):  
Congruence Subgroups ◽  
Hecke Groups

scholarly journals Principal Congruence Subgroups of the Hecke Groups

2000 ◽  
Vol 85 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Mong-Lung Lang ◽  
Chong-Hai Lim ◽  
Ser-Peow Tan
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups

Normal Congruence Subgroups of Hecke Groups

2008 ◽  
Vol 15 (04) ◽  
pp. 707-720
Author(s):  
Bo Chen ◽  
Pingzhi Yuan
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Finite Index ◽  
Principal Congruence ◽  
Important Class ◽  
Congruence Subgroups ◽  
Discrete Subgroups ◽  
Hecke Groups ◽  
Hecke Group ◽  
Normal Congruence

Hecke groups are an important class of discrete subgroups of PSL(2, ℝ), which play an important role in the study of Dirichlet series. Subgroups with finite index of a Hecke group, which are called congruence subgroups, are often used. Let q be a positive integer with [Formula: see text]. For the Hecke group [Formula: see text], the structures of principal congruence subgroups and normal congruence subgroups of level m are investigated in many papers, where m is a prime or a power of an odd prime. In this paper, we deal with the case that the level m is a power of 2.

Principal congruence subgroups of the Hecke groups and related results

2009 ◽  
Vol 40 (4) ◽  
pp. 479-494 ◽  
Author(s):  
Sebahattin Ikikardes ◽  
Recep Sahin ◽  
I. Naci Cangul
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups

Congruence subgroups of Hecke groups

2009 ◽  
Vol 25 (6) ◽  
pp. 931-944 ◽  
Author(s):  
Bo Chen ◽  
Ping Zhi Yuan
Keyword(s):  
Congruence Subgroups ◽  
Hecke Groups

scholarly journals Regular maps and principal congruence subgroups of Hecke groups

2005 ◽  
Vol 26 (3-4) ◽  
pp. 437-456 ◽  
Author(s):  
Ioannis Ivrissimtzis ◽  
David Singerman
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups ◽  
Hecke Groups ◽  
Regular Maps

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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