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.

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.

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 Rankin-Selberg method for Siegel cusp forms

1990 ◽  
Vol 120 ◽  
pp. 35-49 ◽  
Author(s):  
Tadashi Yamazaki
Keyword(s):  
Holomorphic Function ◽  
Half Plane ◽  
Cusp Form ◽  
Functional Equations ◽  
Modular Group ◽  
Cusp Forms ◽  
The Real ◽  
Upper Half Plane ◽  
Siegel Cusp Form

Let Gn (resp. Γn) be the real symplectic (resp. Siegel modular) group of degree n. The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.

scholarly journals On construction of holomorphic cusp forms of half integral weight

1975 ◽  
Vol 58 ◽  
pp. 83-126 ◽  
Author(s):  
Takuro Shintani
Keyword(s):  
Half Plane ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Holomorphic Cusp Forms ◽  
Upper Half Plane

In [10], G.Shimura gave a method of constructing holomorphic cusp forms of even integral weight from given forms of half integral weight. In this paper, we try to present an inverse construction. To state our main result, some notational preliminaries are necessary. We denote by the complex upper half plane.

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.

scholarly journals On monodromy map

1993 ◽  
Vol 16 (4) ◽  
pp. 695-708
Author(s):  
Jharna D. Sengupta
Keyword(s):  
Vector Bundle ◽  
Conjugacy Class ◽  
Equivalence Class ◽  
Half Plane ◽  
Fuchsian Group ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Projective Structures ◽  
Upper Half Plane ◽  
Monodromy Map

LetΓbe a Fuchsian group acting on the upper half-planeUand having signature{p,n,0;v1,v2,…,vn};2p−2+∑j=1n(1−1vj)>0.LetT(Γ)be the Teichmüller space ofΓ. Then there exists a vector bundleℬ(T(Γ))of rank3p−3+noverT(Γ)whose fibre over a pointt∈T(Γ)representingΓtis the space of bounded quratic differentialsB2(Γt)forΓt. LetHom(Γ,G)be the set of all homomorphisms fromΓinto the Mbius groupG.For a given(t,ϕ)∈ℬ(T(Γ))we get an equivalence class of projective structures and a conjugacy class of a homomorphismx∈Hom(Γ,G). Therefore there is a well defined mapΦ:ℬ(T(Γ))→Hom(Γ,G)/G,Φis called the monodromy map. We prove that the monromy map is hommorphism. The casen=0gives the previously known result by Earle, Hejhal Hubbard.

scholarly journals Basis of quadratic differentials for Riemann surfaces with automorphisms

1997 ◽  
Vol 39 (2) ◽  
pp. 193-210
Author(s):  
Gonzalo Riera
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Fundamental Region ◽  
Quadratic Differentials ◽  
Uniformization Theorem ◽  
Group A ◽  
Upper Half Plane

The uniformization theorem says that any compact Riemann surface S of genus g≥2 can be represented as the quotient of the upper half plane by the action of a Fuchsian group A with a compact fundamental region Δ.

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