scholarly journals A remark on conductor, depth and principal congruence subgroups

2021 ◽  
Author(s):  
Michitaka Miyauchi ◽  
Takuya Yamauchi
Keyword(s):  
Principal Congruence ◽  
Congruence Subgroups

APPROXIMATION OF -ANALYTIC TORSION FOR ARITHMETIC QUOTIENTS OF THE SYMMETRIC SPACE

2018 ◽  
Vol 19 (2) ◽  
pp. 307-350
Author(s):  
Jasmin Matz ◽  
Werner Müller
Keyword(s):  
Symmetric Space ◽  
Principal Congruence ◽  
Analytic Torsion ◽  
Congruence Subgroups ◽  
Arithmetic Subgroup ◽  
Limiting Behavior

In [31] we defined a regularized analytic torsion for quotients of the symmetric space $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^{2}$-analytic torsion.

Spectral Estimates for Towers of Noncompact Quotients

1999 ◽  
Vol 51 (2) ◽  
pp. 266-293 ◽  
Author(s):  
Anton Deitmar ◽  
Werner Hoffman
Keyword(s):  
Analogous Result ◽  
Congruence Subgroup ◽  
Kernel Method ◽  
Linear Algebraic Group ◽  
Fixed Number ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Spectral Estimates ◽  
The Family ◽  
Upper Estimate

AbstractWe prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ.

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.

scholarly journals On Conjugacy Classes of Congruence Subgroups of PSL(2, R)

2009 ◽  
Vol 12 ◽  
pp. 264-274 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Congruence Subgroup ◽  
Principal Congruence ◽  
Conjugacy Classes ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup ◽  
Genus One

AbstractLet G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup /overline Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

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