Automorphic functions on the upper half plane, especially modular functions

1968 ◽  
pp. 2-8
Author(s):  
Goro Shimura
Keyword(s):  
Half Plane ◽  
Modular Functions ◽  
Upper Half Plane ◽  
Automorphic Functions

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 On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

scholarly journals Revisiting the Siegel upper half plane II

2004 ◽  
Vol 376 ◽  
pp. 45-67 ◽  
Author(s):  
Pedro J. Freitas ◽  
Shmuel Friedland
Keyword(s):  
Half Plane ◽  
Upper Half Plane

An AF Algebra Associated with the Farey Tessellation

2008 ◽  
Vol 60 (5) ◽  
pp. 975-1000 ◽  
Author(s):  
Florin P. Boca
Keyword(s):  
Half Plane ◽  
Path Algebra ◽  
Cutting Sequences ◽  
Upper Half Plane ◽  
Af Algebras ◽  
Algebra Model

AbstractWe associate with the Farey tessellation of the upper half-plane an AF algebra encoding the “cutting sequences” that define vertical geodesics. The Effros–Shen AF algebras arise as quotients of . Using the path algebra model for AF algebras we construct, for each τ ∈ ( 0, ¼], projections (En) in such that EnEn±1En ≤ τ En.

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