scholarly journals Sup-norm bounds for Eisenstein series

2017 ◽  
Vol 29 (6) ◽  
Author(s):  
Bingrong Huang ◽  
Zhao Xu
Keyword(s):  
Lower Bound ◽  
Eisenstein Series ◽  
Half Plane ◽  
Spectral Parameter ◽  
Geometric Method ◽  
Analytic Method ◽  
Counting Problem ◽  
Efficient Treatment ◽  
Upper Half Plane

AbstractThe paper deals with establishing bounds for Eisenstein series on congruence quotients of the upper half plane, with control of both the spectral parameter and the level. The key observation in this work is that we exploit better the structure of the amplifier by just supporting on primes for the Eisenstein series, which can use both the analytic method as Young did to get a lower bound for the amplifier and the geometric method as Harcos–Templier did to obtain a more efficient treatment for the counting problem.

scholarly journals A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

2021 ◽  
pp. 17-26
Author(s):  
Р.Т. Зуннунов
Keyword(s):  
Half Plane ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Nonlocal Problem ◽  
Existence Of A Solution ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Upper Half Plane ◽  
Generalized Tricomi Equation

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.

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 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.

DIFFERENTIAL ACTIONS ON THE ASYMPTOTIC EXPANSIONS OF NON-HOLOMORPHIC EISENSTEIN SERIES

2009 ◽  
Vol 05 (06) ◽  
pp. 1061-1088 ◽  
Author(s):  
MASANORI KATSURADA ◽  
TAKUMI NODA
Keyword(s):  
Asymptotic Expansion ◽  
Eisenstein Series ◽  
Functional Properties ◽  
Half Plane ◽  
Asymptotic Expansions ◽  
Weight Change ◽  
Complete Asymptotic Expansion ◽  
Upper Half Plane

Let k be an arbitrary even integer, and Ek(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL2(ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of Ek(s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E0(s;z) (due to the first author [16]) to that of Ek(s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on Ek(s;z), including its functional properties (Corollaries 2.1–2.3), its relevant specific values (Corollaries 2.4–2.7), and its asymptotic aspects as z → 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian ΔH,k (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for Ek(s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2).

scholarly journals On the Eisenstein Series for the Principal Congruence Subgroups

1969 ◽  
Vol 34 ◽  
pp. 129-142 ◽  
Author(s):  
Akio Orihara
Keyword(s):  
Eisenstein Series ◽  
Finite Type ◽  
Half Plane ◽  
Fuchsian Group ◽  
Principal Congruence ◽  
Congruence Subgroups ◽  
Upper Half Plane

Let Γ be a Fuchsian group (of finite type) acting on the upper half plane. To each parabolic cusp Ki (i = 1, …, h), corresponds a Eisenstein serie

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.

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