A remark on symplectic matrices
type: Item
instance of: scholarly article
title: A remark on symplectic matrices
zbMATH Open document ID: 0855.11022
DOI: 10.1007/BF02953330
author: Winfried Kohnen
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1996-06-19T00:00:00Z
review text: Let \(\Gamma_g\) be the Siegel modular group of degree \(g\) with the standard subgroup \(\Gamma_{g, \infty}= \{{{*\;*} \choose {0\;*}}\in \Gamma_g\}\). \(\Gamma_g\) acts on the Siegel half-space \({\mathcal H}_g\) in the usual way. If \(h(Z):= \text{det} (\text{Im Z})\), \(Z\in {\mathcal H}_g\), denotes the height, the author investigates the arithmetic function \[ \alpha_Z (n):= \# \Bigl\{ \Gamma_{g, \infty} M;\;M\in \Gamma_g,\;{\textstyle {1\over {n+1}}} \leq h(M \langle Z\rangle)< {\textstyle {1\over n}} \Bigr\}, \] which plays a decisive role in the proof of convergence of the Siegel-Eisenstein series. The author's main result is \[ \alpha_Z (n)= O(n^{(g+ 1)/2- \delta_g+ \varepsilon}), \quad \varepsilon >0, \quad \delta_g= 1\Bigl/ \Bigl( 1+2 \bigl[ {\textstyle {g\over 2}} \bigr]+ {\textstyle {1\over {g+1}}} \Bigr). \] The proof uses analytic properties of Eisenstein series and a version of Landau's theorem.
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 878381
zbMATH Keywords: symplectic matrices
zbMATH Keywords: Siegel modular group
zbMATH Keywords: height
zbMATH Keywords: arithmetic function
zbMATH Keywords: Siegel-Eisenstein series
zbMATH Keywords: Landau's theorem
reviewed by: Aloys Krieg
MaRDI profile type: Publication
cites work: Konvergenz verallgemeinerter Eisensteinscher Reihen
cites work: Q3950619
cites work: EISENSTEIN SERIES ON THE SYMPLECTIC GROUP
cites work: On zeta functions associated with prehomogeneous vector spaces
cites work: Q3909097
full work available at URL: https://doi.org/10.1007/bf02953330
OpenAlex ID: W2006738894
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description: scientific article; zbMATH DE number 878381