scholarly journals On the Eisenstein Series for the Principal Congruence Subgroups

1969 ◽  
Vol 34 ◽  
pp. 129-142 ◽  
Author(s):  
Akio Orihara
Keyword(s):  
Eisenstein Series ◽  
Finite Type ◽  
Half Plane ◽  
Fuchsian Group ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Upper Half Plane

Let Γ be a Fuchsian group (of finite type) acting on the upper half plane. To each parabolic cusp Ki (i = 1, …, h), corresponds a Eisenstein serie

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.

The Values of Modular Functions and Modular Forms

2006 ◽  
Vol 49 (4) ◽  
pp. 526-535 ◽  
Author(s):  
So Young Choi
Keyword(s):  
Modular Form ◽  
Half Plane ◽  
Finite Index ◽  
Modular Forms ◽  
Fuchsian Group ◽  
Modular Functions ◽  
Genus Zero ◽  
Recursive Formulas ◽  
Upper Half Plane ◽  
Recursive Relations

AbstractLet Γ0 be a Fuchsian group of the first kind of genus zero and Γ be a subgroup of Γ0 of finite index of genus zero. We find universal recursive relations giving the qr-series coefficients of j0 by using those of the qhs -series of j, where j is the canonical Hauptmodul for Γ and j0 is a Hauptmodul for Γ0 without zeros on the complex upper half plane (here qℓ := e2πiz/ℓ). We find universal recursive formulas for q-series coefficients of any modular form on in terms of those of the canonical Hauptmodul .

Transitivity for the modular group

1980 ◽  
Vol 88 (3) ◽  
pp. 409-423 ◽  
Author(s):  
Mark Sheingorn
Keyword(s):  
Real Number ◽  
Half Plane ◽  
Limit Point ◽  
Fuchsian Group ◽  
Modular Group ◽  
Fundamental Region ◽  
Hyperbolic Line ◽  
Upper Half Plane

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.

scholarly journals Automorphic Schwarzian equations

2020 ◽  
Vol 32 (6) ◽  
pp. 1621-1636
Author(s):  
Abdellah Sebbar ◽  
Hicham Saber
Keyword(s):  
Differential Equation ◽  
Representation Theory ◽  
Eisenstein Series ◽  
Half Plane ◽  
Index Subgroup ◽  
Modular Functions ◽  
Upper Half Plane ◽  
Finite Index Subgroup ◽  
Equivariant Functions ◽  
Do So

AbstractThis paper concerns the study of the Schwartz differential equation {\{h,\tau\}=s\operatorname{E}_{4}(\tau)}, where {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of {\operatorname{SL}_{2}({\mathbb{Z}})}. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of {\operatorname{SL}_{2}({\mathbb{Z}})}. This also leads to the solutions to the Fuchsian differential equation {y^{\prime\prime}+s\operatorname{E}_{4}y=0}.

scholarly journals On boundaries of Schottky spaces

1976 ◽  
Vol 62 ◽  
pp. 97-124 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Compact Riemann Surface ◽  
Teichmüller Space ◽  
Schottky Group ◽  
Teichmuller Space ◽  
Upper Half Plane

Let S be a compact Riemann surface and let Sn be the surface obtained from S in the course of a pinching deformation. We denote by Γn the quasi-Fuchsian group representing Sn in the Teichmüller space T(Γ), where Γ is a Fuchsian group with U/Γ = S (U: the upper half plane). Then in the previous paper [7] we showed that the limit of the sequence of Γn is a cusp on the boundary ∂T(Γ). In this paper we will consider the case of Schottky space . Let Gn be a Schottky group with Ω(Gn)/Gn = Sn. Then the purpose of this paper is to show what the limit of Gn is.

Fuchsian Embeddings in the Bianchi Groups

1987 ◽  
Vol 39 (6) ◽  
pp. 1434-1445 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Half Plane ◽  
Modular Group ◽  
Function Theory ◽  
Automorphic Function ◽  
Congruence Subgroups ◽  
Ring Of Integers ◽  
Linear Fractional Transformations ◽  
Upper Half Plane ◽  
Total Structure ◽  
Image Position

If d is a positive square free integer we let Od be the ring of integers in and we let Γd = PSL2(Od), the group of linear fractional transformationsand entries from Od {if d = 1, ad – bc = ±1}. The Γd are called collectively the Bianchi groups and have been studied extensively both as abstract groups and in automorphic function theory {see references}. Of particular interest has been Γ1 – the Picard group. Group theoretically Γ1, is very similar to the classical modular group M = PSL2(Z) both in its total structure [4, 6], and in the structure of its congruence subgroups [8]. Where Γ1 and M differ greatly is in their action on the complex place C. M is Fuchsian and therefore acts discontinuously in the upper half-plane and every subgroup has the same property.

scholarly journals Mixed cusp forms and parabolic cohomology

1997 ◽  
Vol 62 (3) ◽  
pp. 279-289
Author(s):  
Min Ho Lee
Keyword(s):  
Half Plane ◽  
Fuchsian Group ◽  
Cusp Forms ◽  
Holomorphic Map ◽  
Cohomology Space ◽  
Upper Half Plane

AbstractLet Sk, l(Γ, ω, χ) be the space of mixed cusp forms of type (k, l) associated to a Fuchsian group Γ, a holomorphic map ω: ℋ → ℋ of the upper half plane into itself and a homomorphism χ: Γ → SL(2, R) such that ω and χ are equivariant. We construct a map from Sk, l(Γ, ω, χ) to the parabolic cohomology space of Γ with coefficients in some Γ-module and prove that this map is injective.

scholarly journals Fourier expansions of complex-valued Eisenstein series on finite upper half planes

2006 ◽  
Vol 2006 ◽  
pp. 1-17 ◽  
Author(s):  
Anthony Shaheen ◽  
Audrey Terras
Keyword(s):  
Eisenstein Series ◽  
Half Plane ◽  
Modular Forms ◽  
Bessel Functions ◽  
Kloosterman Sums ◽  
Fourier Expansions ◽  
Wave Forms ◽  
Upper Half Plane ◽  
Complex Valued

We consider complex-valued modular forms on finite upper half planesHqand obtain Fourier expansions of Eisenstein series invariant under the groupsΓ=SL(2,Fp)andGL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincaré upper half plane —theK-Bessel functions being replaced by Kloosterman sums.

A note on cusp forms and representations of SL2(𝔽𝑝)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhe Chen
Keyword(s):  
Finite Field ◽  
Half Plane ◽  
Congruence Subgroup ◽  
Holomorphic Functions ◽  
Principal Congruence ◽  
Cusp Forms ◽  
Cohomology Space ◽  
Principal Congruence Subgroup ◽  
Upper Half Plane ◽  
Space Of Cusp Forms

AbstractCusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup \Gamma(p), 𝑝 a prime, is acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). In this note, we compute the relation between these two spaces in the weight 2 case.

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