Zeta-functions of binary Hermitian forms and special values of Eisenstein series
type: Item
instance of: scholarly article
title: Zeta-functions of binary Hermitian forms and special values of Eisenstein series
zbMATH Open document ID: 0659.10021
DOI: 10.1007/BF01457865
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164273
review text: The work under review is a systematic investigation of zeta-functions of binary Hermitian forms over the ring \(\mathfrak o\) of integers of an imaginary quadratic number field \(K={\mathbb{Q}}(\sqrt{D})\) with discriminant \(D<0\). First we develop a theory of representations of numbers by binary Hermitian forms over \(\mathfrak o\) which parallels the classical theory of representations of integers by binary quadratic forms over \(\mathbb{Z}.\) A key result is a bijection described in Theorem 2.3. This bijection means that the sum of the numbers of representations (modulo \(\mathrm{SL}(2,\mathfrak o)\)-units) of an integer \(k\neq 0\) by a set of representatives of the \(\mathrm{SL}(2,\mathfrak o)\)-classes of binary Hermitian forms over \({\mathfrak o}\) with discriminant \(\Delta\) is equal to the number of cosets \(\lambda +k{\mathfrak o}\) with \(\lambda\in {\mathfrak o}\), \(\lambda \bar{\lambda}+\Delta \equiv 0 \bmod k\). The number of these cosets is explicitly computed, and the result is applied to the study of zeta-functions of (definite or indefinite) Hermitian forms. For simplicity we restrict in this review to the case of positive definite forms over \(\mathfrak o\). Let \(f\) be a positive definite binary Hermitian form over \(\mathfrak o\) with \(\mathrm{SL}(2,\mathfrak o)\)-unit group \(\mathcal E_ 1(f)\) and define \[ Z_{\mathfrak o}(f,s):=\frac{1}{| {\mathcal E}_ 1(f)|}\sum_{u,v\in \mathfrak o, <u,v>=\mathfrak o}f(u,v)^{-1-s},\quad Z(\Delta,s):=\sum_{[f]}Z_{\mathfrak o}(f,s), \] where the latter sum extends over a representative system of the \(\mathrm{SL}(2,\mathfrak o)\)-orbits of positive definite binary Hermitian forms over \(\mathfrak o\) with discriminant \(\Delta\). Then the analytic version of the results from above says that \[ Z(\Delta,s)=\theta (\Delta,s) \zeta (s) L(s+1,\chi_ D)^{-1}, \] where \(\theta (\Delta,s)\) is a finite Euler product whose factors are explicitly known. Second we consider the genus \(\mathcal G(f)\) of the positi…
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 4077330
zbMATH Keywords: zeta-functions of binary Hermitian forms
zbMATH Keywords: representations of numbers
zbMATH Keywords: genus
zbMATH Keywords: Dirichlet series
zbMATH Keywords: local densities
zbMATH Keywords: Hilbert symbol
zbMATH Keywords: Eisenstein series
zbMATH Keywords: tables
zbMATH Keywords: special values of Eisenstein series
zbMATH Keywords: representation numbers
zbMATH Keywords: Fourier coefficients of cusp forms of weight 2
zbMATH Keywords: L-series of certain elliptic curves
zbMATH Keywords: periods
author: Jürgen Elstrodt
author: Fritz Grunewald
author: Jens Mennicke
MaRDI profile type: Publication
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OpenAlex ID: W1998580304
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description: scientific article; zbMATH DE number 4077330