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.

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.

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].

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 Dedekind sums for a fuchsian group, II

1974 ◽  
Vol 53 ◽  
pp. 171-187 ◽  
Author(s):  
Larry Joel Goldstein
Keyword(s):  
Fuchsian Group ◽  
Modular Group ◽  
Preceding Paper ◽  
Principal Congruence ◽  
Dedekind Sums ◽  
Natural Question ◽  
Congruence Subgroups ◽  
Limit Formula ◽  
Upper Half Plane ◽  
The Subject

In [1] we derived a generalization of Kronecker’s first limit formula. Our generalization was a limit formula for the Eisenstein series for an arbitrary cusp of a Fuchsian group Γ of the first kind operating on the complex upper half-plane H. In that work, we introduced Dedekind sums associated to the principal congruence subgroups Γ(N) of the elliptic modular group. The work of our preceding paper suggests a natural question: Is there a generalization of Kronecker’s second limit formula to the setting of a general Fuchsian group of the first kind? The answer to this question is the subject of this paper.

scholarly journals Subgroups of infinite index in the modular group

1978 ◽  
Vol 19 (1) ◽  
pp. 33-43 ◽  
Author(s):  
W. W. Stothers
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Modular Group ◽  
Infinite Index ◽  
Upper Half Plane ◽  
Extended Complex ◽  
Extended Complex Plane

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

scholarly journals Rational period functions of the modular group II

1981 ◽  
Vol 22 (2) ◽  
pp. 185-197 ◽  
Author(s):  
Marvin I. Knopp
Keyword(s):  
Half Plane ◽  
Functional Equations ◽  
Modular Group ◽  
Fourier Expansion ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Linear Fractional Transformations ◽  
Upper Half Plane ◽  
Image Position ◽  
Rational Period Functions

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the formwhere k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

Dedekind sums and Hecke operators

1980 ◽  
Vol 88 (1) ◽  
pp. 11-14 ◽  
Author(s):  
L. Alayne Parson
Keyword(s):  
Elementary Proof ◽  
Hecke Operators ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Dedekind Eta Function ◽  
Dedekind Sum ◽  
Modular Transformation ◽  
Special Case

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

Sign in / Sign up

Export Citation Format

Share Document