On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight
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 finite sum of products of \(L\)-functions. There are results on analogues of Salié sums which are of independent interest.
scientific article; zbMATH DE number 5378834