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}.

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 A CONVENIENT COORDINATIZATION OF SIEGEL–JACOBI DOMAINS

2012 ◽  
Vol 24 (10) ◽  
pp. 1250024 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Half Plane ◽  
Order Differential Equation ◽  
Quantum Evolution ◽  
Classical Motion ◽  
Matrix Riccati Equation ◽  
First Order ◽  
Upper Half Plane ◽  
Linear Differential

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball [Formula: see text] as the sum of the Kähler two-form on ℂn and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a hermitian linear Hamiltonian in the generators of the Jacobi group [Formula: see text] are described by a matrix Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂn, with coefficients depending also on [Formula: see text]. Hn denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on [Formula: see text]. When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable [Formula: see text] becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane [Formula: see text], where [Formula: see text] denotes the Siegel upper half plane.

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 Elliptic Zeta Functions and Equivariant Functions

2018 ◽  
Vol 61 (2) ◽  
pp. 376-389 ◽  
Author(s):  
Abdellah Sebbar ◽  
Isra Al-Shbeil
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Zeta Functions ◽  
Close Connection ◽  
Modular Subgroup ◽  
Upper Half Plane ◽  
Equivariant Functions

AbstractIn this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing theWeierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

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.

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.

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

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