The Chowla-Selberg Method for Fourier Expansion of Higher Rank Eisenstein Series

1985 ◽  
Vol 28 (3) ◽  
pp. 280-294 ◽  
Author(s):  
Audrey Terras
Keyword(s):  
Eisenstein Series ◽  
Fourier Expansion ◽  
Bessel Functions ◽  
Maximal Rank ◽  
Hecke Operators ◽  
Divisor Functions ◽  
Fourier Expansions ◽  
Higher Rank

AbstractThe terms of maximal rank in Fourier expansions of Eisenstein series for GL(n, ℤ) are obtained by an analogue of a method of Chowla and Selberg. The coefficients involve matrix analogues of divisor functions as well as K-Bessel functions for GL(n). The discussion involves a few properties of Hecke operators.

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 On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

THE MINIMAL MODULAR FORM ON QUATERNIONIC

2020 ◽  
pp. 1-34
Author(s):  
Aaron Pollack
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Explicit Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Reductive Group ◽  
Automorphic Function ◽  
Unipotent Radical ◽  
Exceptional Groups ◽  
Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

Fourier expansions and multivariable bessel functions concerning radiation problems

1996 ◽  
Vol 47 (2) ◽  
pp. 183-189 ◽  
Author(s):  
G. Dattoli ◽  
C. Chiccoli ◽  
S. Lorenzutta ◽  
G. Maino ◽  
M. Richetta ◽  
...  
Keyword(s):  
Bessel Functions ◽  
Fourier Expansions

Infinite-variable Bessel functions of the Anger type and the Fourier expansions

1997 ◽  
Vol 39 (2) ◽  
pp. 163-176 ◽  
Author(s):  
S. Lorenzutta ◽  
G. Maino ◽  
G. Dattoli ◽  
A. Torre ◽  
C. Chiccoli
Keyword(s):  
Bessel Functions ◽  
Fourier Expansions

scholarly journals Dirichlet series in the theory of Siegel modular forms

1984 ◽  
Vol 95 ◽  
pp. 73-84 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Dirichlet Series ◽  
Eisenstein Series ◽  
Half Space ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Siegel Modular Forms

We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

scholarly journals On the Fourier Expansions of Eisenstein Series of Some Types

2006 ◽  
Vol 13 (1) ◽  
pp. 55-78
Author(s):  
Nikoloz Kachakhidze
Keyword(s):  
Eisenstein Series ◽  
Dirichlet Character ◽  
Fourier Expansions

Abstract Bases of the spaces of Eisenstein series 𝐸𝑘(Γ0(4𝑁), χ) (𝑘 ∈ ℕ, 𝑘 ≥ 3, 𝑁 is an odd natural and square-free) and , ℕ is an odd natural and square-free) are constructed for any Dirichlet character mod 4𝑁 and Fourier expansions of these series are obtained.

scholarly journals K-Bessel functions in two variables

2003 ◽  
Vol 2003 (14) ◽  
pp. 909-916
Author(s):  
Hacen Dib
Keyword(s):  
Bessel Function ◽  
Linear Combination ◽  
Explicit Formula ◽  
Bessel Functions ◽  
Higher Rank

The Bessel-Muirhead hypergeometric system (or0F1-system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients, in order to express theK-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed.

scholarly journals Hecke operators on half-integral weight Siegel Eisenstein series

2017 ◽  
Vol 13 (09) ◽  
pp. 2335-2372
Author(s):  
Lynne H. Walling
Keyword(s):  
Eisenstein Series ◽  
Hecke Operators ◽  
Linear Combinations ◽  
Half Integral Weight ◽  
Multiplicity One ◽  
Integral Weight

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level [Formula: see text] where [Formula: see text] is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of [Formula: see text], commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of [Formula: see text], obtaining a multiplicity-one result.

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