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].

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 .

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.

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

2006 ◽  
Vol 16 (04) ◽  
pp. 911-923 ◽  
Author(s):  
S. P. BANKS ◽  
SONG YI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Half Plane ◽  
Complex Plane ◽  
Fundamental Domain ◽  
Function Theory ◽  
Theta Series ◽  
The Hyperbolic Metric ◽  
Upper Half Plane ◽  
Definition Of

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

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 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.

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 Transformation of Some Lambert Series and Cotangent Sums

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 840
Author(s):  
Namhoon Kim
Keyword(s):  
Half Plane ◽  
Modular Group ◽  
Transformation Formula ◽  
Contour Integral ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Simple Derivation ◽  
Special Cases ◽  
Upper Half Plane

By considering a contour integral of a cotangent sum, we give a simple derivation of a transformation formula of the series A ( τ , s ) = ∑ n = 1 ∞ σ s − 1 ( n ) e 2 π i n τ for complex s under the action of the modular group on τ in the upper half plane. Some special cases directly give expressions of generalized Dedekind sums as cotangent sums.

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).

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