Modular forms on the upper half-plane

1980 ◽  
pp. 177-196
Author(s):  
Gérard Lion ◽  
Michèle Vergne
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Upper Half Plane

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 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).

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 Elliptic Zeta Functions and Equivariant Functions

2018 ◽  
Vol 61 (2) ◽  
pp. 376-389 ◽  
Author(s):  
Abdellah Sebbar ◽  
Isra Al-Shbeil
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Zeta Functions ◽  
Close Connection ◽  
Modular Subgroup ◽  
Upper Half Plane ◽  
Equivariant Functions

AbstractIn this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing theWeierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

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).

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