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 On the Schwarz lemma at the upper half plane

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2995-3011
Author(s):  
Bülent Örnek
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Half Plane ◽  
Simple Proof ◽  
Boundary Point ◽  
Schwarz Lemma ◽  
The Real ◽  
Upper Half Plane ◽  
Boundary Schwarz Lemma ◽  
The Schwarz Lemma

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f(z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), |f'(0)| by assuming that the f(z) function is also holomorphic at the boundary point z = 0 on the real axis with f(0)=Rf(i).

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.

scholarly journals Fourier coefficients of Siegel cusp forms of degree two

1984 ◽  
Vol 93 ◽  
pp. 149-171 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

Our purpose is to prove the followingTheorem. Let k be an even integer ≥ 6. Letbe a Siegel cusp form of degree two, weight k. Then we have

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

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 On construction of holomorphic cusp forms of half integral weight

1975 ◽  
Vol 58 ◽  
pp. 83-126 ◽  
Author(s):  
Takuro Shintani
Keyword(s):  
Half Plane ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Holomorphic Cusp Forms ◽  
Upper Half Plane

In [10], G.Shimura gave a method of constructing holomorphic cusp forms of even integral weight from given forms of half integral weight. In this paper, we try to present an inverse construction. To state our main result, some notational preliminaries are necessary. We denote by the complex upper half plane.

scholarly journals Estimates for fourier coefficients of siegel cusp forms of degree two, II

1992 ◽  
Vol 128 ◽  
pp. 171-176 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Positive Definite ◽  
Cusp Forms ◽  
Integral Weight ◽  
Siegel Cusp Form

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that(1)(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain(2)

scholarly journals Joint Universality of the Zeta-Functions of Cusp Forms

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2161
Author(s):  
Renata Macaitienė
Keyword(s):  
Cusp Form ◽  
Modular Group ◽  
Zeta Functions ◽  
Rational Numbers ◽  
Cusp Forms ◽  
Algebraic Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Linearly Independent ◽  
Joint Universality Theorem

Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ(s,F) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts (ζ(s+ih1τ,F),⋯,ζ(s+ihrτ,F)) is proved. Here, h1,⋯,hr are algebraic numbers linearly independent over the field of rational numbers.

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].

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