On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight
type: Item
instance of: scholarly article
title: On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight
zbMATH Open document ID: 1167.11018
author: Yoshinori Mizuno
published in: Journal of Number Theory
publication date: 2008-12-05T00:00:00Z
review text: There are several known theorems of the following basic structure: Certain averages of automorphic forms evaluated at suitable points (e.g. CM points) give rise to Fourier coefficients of another class of automorphic forms. The paper under review gives another contribution to this topic: Let \(K\) be an imaginary quadratic number field of discriminant \(-D\) and ring of integers \({\mathcal O}\). For any function \(f\) on the three-dimensional hyperbolic space \(\mathbb H^3\) which is automorphic with respect to \(\text{SL}_2({\mathcal O})\) the author defines an average value \(\text{Tr}_m(f)\) over the three-dimensional analogue of CM points depending on a natural number \(m\). If \(f\) is an Eisenstein series on \(\mathbb H^3\) for the group \(\text{SL}_2({\mathcal O})\) then \(\text{Tr}_m(f)\) is essentially equal to the Fourier coefficient of a linear combination of Eisenstein series on the upper half-plane \(\mathbb H^2\) for the group \(\Gamma_0(D)\) (see Theorems 1 and 5). An analogous result holds for the Poincaré series constructed with the Bessel functions \(I_\nu\) instead of the Eisenstein series on \(\mathbb H^3\) and \(\mathbb H^2\), respectively. (The author calls these series Niebur Poincaré series since they appear in a paper of \textit{D. Niebur} [Nagoya Math. J. 52, 133--145 (1973; Zbl 0288.10010)]. The reviewer wants to point out that the same series came up simultaneously in the Ph. D. Thesis of \textit{H. Neunhöffer} [Über die analytische Fortsetzung von Poincaré-Reihen. Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturwiss. Kl. 1973, 2. Abhdl. 62 S. (1973; Zbl 0272.10015)].) In the course of the proof of Theorem 1 a simple arithmetic expression for the zeta function of representation numbers by positive definite integral binary Hermitian forms is given (see Theorem 3). Theorem 4 gives a nice expression for a certain weighted sum of all \(\text{Tr}_m(f)\) \((m\in \mathbb N)\) for an Eisenstein series \(f\) on \(\mathbb H^3\) in terms of a fin…
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 5378834
zbMATH Keywords: Poincaré series
zbMATH Keywords: Eisenstein series
zbMATH Keywords: average value
zbMATH Keywords: CM point
zbMATH Keywords: three-dimensional hyperbolic space
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.10.012
OpenAlex ID: W2079898614
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DOI: 10.1016/J.JNT.2007.10.012
description: scientific article; zbMATH DE number 5378834