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.

scholarly journals Rational functions, cotangent sums and Eichler integrals

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Johann Franke
Keyword(s):  
Eisenstein Series ◽  
Half Plane ◽  
Modular Forms ◽  
Holomorphic Functions ◽  
Rational Functions ◽  
Close Connection ◽  
General Transformation ◽  
Fourier Expansions ◽  
Upper Half Plane ◽  
Eichler Integrals

AbstractWith the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.

FOURIER EXPANSIONS WITH MODULAR FORM COEFFICIENTS

2009 ◽  
Vol 05 (08) ◽  
pp. 1433-1446 ◽  
Author(s):  
AHMAD EL-GUINDY
Keyword(s):  
Fixed Point ◽  
Modular Form ◽  
Half Plane ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Genus Zero ◽  
Modular Curve ◽  
Hyperelliptic Involution ◽  
Fourier Expansions ◽  
Upper Half Plane

In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X0(l) is hyperelliptic (with hyperelliptic involution of the Atkin–Lehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X0(l) has genus zero).

RATIONAL EQUIVARIANT FORMS

2012 ◽  
Vol 08 (04) ◽  
pp. 963-981 ◽  
Author(s):  
ABDELKRIM EL BASRAOUI ◽  
ABDELLAH SEBBAR
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Discrete Group ◽  
Genus Zero ◽  
Modular Subgroup ◽  
Upper Half Plane ◽  
Schwarz Derivative

We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.

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 .

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 Non-Existence of Ramanujan Congruences in Modular Forms of Level Four

2011 ◽  
Vol 63 (6) ◽  
pp. 1284-1306 ◽  
Author(s):  
Michael Dewar
Keyword(s):  
Partition Function ◽  
Modular Form ◽  
Half Plane ◽  
Modular Forms ◽  
Generating Functions ◽  
Open Questions ◽  
Ramanujan Congruences ◽  
Upper Half Plane

AbstractRamanujan famously found congruences like p(5n+4) ≡ 0 mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Г1(4) that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.

ZERO DISTRIBUTION AND DECAY AT INFINITY OF DRINFELD MODULAR COEFFICIENT FORMS

2011 ◽  
Vol 07 (03) ◽  
pp. 671-693 ◽  
Author(s):  
ERNST-ULRICH GEKELER
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Fundamental Domain ◽  
Modular Group ◽  
Zero Distribution ◽  
Upper Half Plane

Let Γ = GL (2, 𝔽q[T]) be the Drinfeld modular group, which acts on the rigid analytic upper half-plane Ω. We determine the zeroes of the coefficient modular forms aℓk on the standard fundamental domain [Formula: see text] for Γ on Ω, along with the dependence of |aℓk(z)| on [Formula: see text].

scholarly journals Jacobi-like forms and power series bundles

2002 ◽  
Vol 66 (2) ◽  
pp. 301-311 ◽  
Author(s):  
Min Ho Lee
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Half Plane ◽  
Modular Forms ◽  
Vector Bundles ◽  
Discrete Subgroup ◽  
Holomorphic Functions ◽  
Jacobi Forms ◽  
Upper Half Plane ◽  
Formal Power

Jacobi-like forms are certain formal power series which generalise Jacobi forms in some sense, and they are closely linked to modular forms when their coefficients are holomorphic functions on the Poincaré upper half plane. We construct two types of vector bundles whose fibres are isomorphic to the space of certain formal power series and whose sections can be identified with Jacobi-like forms for a discrete subgroup of SL (2,ℝ).

Elementary proofs of relations between Eisenstein series

1977 ◽  
Vol 76 (2) ◽  
pp. 107-117 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Diophantine Equations ◽  
Vector Spaces ◽  
Integral Weight ◽  
Fourier Expansions ◽  
Elementary Methods

SynopsisEisenstein series are entire modular forms Ek of even integral weight k≧ 4 with Fourier expansions given by (1.1). There are numerous identities, such as E8 = , relating these series. These are usually proved by arguments making use of the dimensions of vector spaces of modular forms, and not directly. The paper shows how such identities can be proved by elementary methods by studying chains of solutions of Diophantine equations of the form xξ+yη = n.

Sign in / Sign up

Export Citation Format

Share Document