On the Parabolic Generators of the Principal Congruence Subgroups of the Modular Group

1952 ◽  
Vol 74 (2) ◽  
pp. 435 ◽  
Author(s):  
Emil Grosswald
Keyword(s):  
Modular Group ◽  
Principal Congruence ◽  
Congruence Subgroups

scholarly journals Dedekind sums for a fuchsian group, II

1974 ◽  
Vol 53 ◽  
pp. 171-187 ◽  
Author(s):  
Larry Joel Goldstein
Keyword(s):  
Fuchsian Group ◽  
Modular Group ◽  
Preceding Paper ◽  
Principal Congruence ◽  
Dedekind Sums ◽  
Natural Question ◽  
Congruence Subgroups ◽  
Limit Formula ◽  
Upper Half Plane ◽  
The Subject

In [1] we derived a generalization of Kronecker’s first limit formula. Our generalization was a limit formula for the Eisenstein series for an arbitrary cusp of a Fuchsian group Γ of the first kind operating on the complex upper half-plane H. In that work, we introduced Dedekind sums associated to the principal congruence subgroups Γ(N) of the elliptic modular group. The work of our preceding paper suggests a natural question: Is there a generalization of Kronecker’s second limit formula to the setting of a general Fuchsian group of the first kind? The answer to this question is the subject of this paper.

scholarly journals Modular symmetry at level 6 and a new route towards finite modular groups

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Cai-Chang Li ◽  
Xiang-Gan Liu ◽  
Gui-Jun Ding
Keyword(s):  
Invariant Theory ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Modular Invariant ◽  
Benchmark Model ◽  
Modular Groups ◽  
Principal Congruence Subgroup ◽  
Comprehensive Study

Abstract We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.

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.

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