The Fourier Expansion of Eisenstein Series for GL(3,Z)

1982 ◽  
Vol 273 (2) ◽  
pp. 679 ◽  
Author(s):  
K. Imai ◽  
A. Terras
Keyword(s):  
Eisenstein Series ◽  
Fourier Expansion

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 $\operatorname{GSp}_{2n}$ .

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 Rationality of the Petersson Inner Product of Cohen’s Kernels

2021 ◽  
pp. 1-19
Author(s):  
Yuanyi You ◽  
Yichao Zhang
Keyword(s):  
Eisenstein Series ◽  
Fourier Expansion ◽  
Inner Product ◽  
Double Eisenstein Series

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series [Formula: see text] of Diamantis and O’Sullivan, we prove a formula of the Petersson inner product of Cohen’s kernel and one of its twists, and obtain a rationality result. This extends a result of Kohnen and Zagier.

scholarly journals THE DOUBLE SHUFFLE RELATIONS FOR MULTIPLE EISENSTEIN SERIES

2017 ◽  
Vol 230 ◽  
pp. 180-212 ◽  
Author(s):  
HENRIK BACHMANN ◽  
KOJI TASAKA
Keyword(s):  
Eisenstein Series ◽  
Hopf Algebras ◽  
Fourier Expansion ◽  
Iterated Integrals

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

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.

Polynomials and Reciprocals of Eisenstein series

2020 ◽  
pp. 1-11
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Fourier Coefficients ◽  
Circle Method ◽  
Upper And Lower Bounds ◽  
Easy Access ◽  
Asymptotic Results ◽  
Complex Half Plane ◽  
General Weight

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series [Formula: see text], especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of [Formula: see text] of general weight, and provides some upper and lower bounds for their growth.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

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