PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

2009 ◽  
Vol 05 (05) ◽  
pp. 845-857 ◽  
Author(s):  
MARVIN KNOPP ◽  
GEOFFREY MASON
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

On the algebra of modular forms on a congruence subgroup

1979 ◽  
Vol 86 (3) ◽  
pp. 461-466 ◽  
Author(s):  
A. J. Scholl
Keyword(s):  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Expansion ◽  
Congruence Subgroup ◽  
Complex Numbers ◽  
Graded Algebra

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.

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 .

EICHLER COHOMOLOGY THEOREM FOR VECTOR-VALUED MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1765-1788 ◽  
Author(s):  
JOSE GIMENEZ
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Large Integer ◽  
Eichler Cohomology ◽  
Vector Valued

We prove the Eichler cohomology theorem for vector-valued modular forms of large integer weights on the full modular group.

scholarly journals ON THE MODULUS OF THE SELBERG ZETA-FUNCTIONS IN THE CRITICAL STRIP

2015 ◽  
Vol 20 (6) ◽  
pp. 852-865
Author(s):  
Andrius Grigutis ◽  
Darius Šiaučiūnas
Keyword(s):  
Riemann Surface ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Zeta Functions ◽  
Critical Strip ◽  
The Real ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.

scholarly journals On vector spaces of certain modular forms of given weights

1977 ◽  
Vol 16 (3) ◽  
pp. 371-378 ◽  
Author(s):  
A.R. Aggarwal ◽  
M.K. Agrawal
Keyword(s):  
Vector Space ◽  
Modular Form ◽  
Modular Forms ◽  
Modular Group ◽  
Vector Spaces ◽  
Unique Element ◽  
Lecture Notes ◽  
Countably Infinite

Let p be a rational prime and Qp be the field of p–adic numbers. Jean-Pierre Serre [Lecture Notes in Mathematics, 350, 191–268 (1973)] had defined p–adic modular forms as the limits of sequences of modular forms over the modular group SL2(Z). He proved that with each non-zero p–adic modular form there is associated a unique element called its weight k. The p–adic modular forms having the same weight form a Qp–vector space.The object of this paper is to obtain a basis of p–adic modular forms and thus to know precisely all p–adic modular forms of a given weight k. The dimension of such modular forms as a Qp–vector space is countably infinite.

scholarly journals Fourier Coefficients of Vector-valued Modular Forms of Dimension 2

2014 ◽  
Vol 57 (3) ◽  
pp. 485-494 ◽  
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Dimension 2 ◽  
Vector Valued

AbstractWe prove the following theorem. Suppose that F = ( f1, f2) is a 2-dimensional, vectorvalued modular form on SL2(ℤ) whose component functions f1, f2 have rational Fourier coefficients with bounded denominators. Then f1, f2 are classical modular forms on a congruence subgroup of the modular group.

scholarly journals On Generalized Modular forms and their Applications

2008 ◽  
Vol 192 ◽  
pp. 119-136 ◽  
Author(s):  
Winfried Kohnen ◽  
Geoffrey Mason
Keyword(s):  
Finite Index ◽  
Modular Forms ◽  
Vertex Operator ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Integral Weight

AbstractWe study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

scholarly journals Theta functions and modular jets

1978 ◽  
Vol 72 ◽  
pp. 83-92 ◽  
Author(s):  
H. D. Fegan
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Modular Group ◽  
Theta Functions ◽  
Line Bundles ◽  
Canonical Bundle ◽  
Jet Bundle ◽  
Upper Half Plane ◽  
Vector Valued Function ◽  
Vector Valued

Let⌈be a subgroup of the modular groupPSL(2,Z) then⌈acts on the upper half planeH={zЄC: Imz> 0} and we can form the Riemann surfaceM=H/⌈, see [3]. The complex line bundles on a Riemann surfaceMform a groupH1(M,*), see [4], and whenever we raise a line bundle to a power it will be in this group. Letκdenote the canonical bundle onMthen a modular form of weightνis a section of the bundle. A modularn-jet is then a section ofJnthen-th jet bundle, see [7]. We can reformulate these ideas in the following terms. A modular form can be viewed as a functionΦ: H→Cand a modularn-jet as a vector valued functionΦ: H → Cn+1both of which satisfy a transformation law under the elements ofΓ.

Weierstrass Points at the Cusps of Γ0(16p) and Hyperellipticity of Γ0(n)

1971 ◽  
Vol 23 (6) ◽  
pp. 960-968 ◽  
Author(s):  
H. Larcher
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Weierstrass Point ◽  
Weierstrass Points ◽  
Linear Fractional Transformations ◽  
Fixed Positive Integer

For a fixed positive integer n we consider the subgroup Γ0(n) of the modular group Γ(l). Γ0(n) consists of all linear fractional transformations L: z → (az + b)/(cz + d) with rational integers a, b, c, d, determinant ad – bc = 1, and c ≡ 0(mod n). If ℋ = {z|z = x + iy, x and y real and y > 0} is the upper half of the z-plane then S0 = S0(n) = ℋ/Γ0(n), properly compactified, is a compact Riemann surface whose genus we denote by g(n). A point P of a Riemann surface S of genus g is called a Weierstrass point if there exists a function on S that has a pole of order α ≦ g at P and is regular everywhere else on S.Lehner and Newman started the search for Weierstrass points of S0 (or, loosely, of Γ0(n)).

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