On certain elliptic conjugacy classes of the Siegel modular group
type: Item
instance of: scholarly article
title: On certain elliptic conjugacy classes of the Siegel modular group
zbMATH Open document ID: 0572.10020
author: Ki-ichiro Hashimoto
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: The author announces some results on the elliptic conjugacy classes in \(\Gamma_ n:=Sp(n,{\mathbb{Z}})\), the Siegel modular group of degree n. Let \(\Phi =\Phi_ m(X)\) be the mth cyclotomic polynomial and denote by \({\mathfrak H}(\Phi)\), for a subgroup \({\mathfrak H}\) of Sp(n,\({\mathbb{C}})\), the set of elements of \({\mathfrak H}\) whose minimal polynomials are \(\Phi\). Then in Theorem 1, the author states a bijective correspondence between the set of conjugacy classes in \(\Gamma_ n(\Phi)\) and the set of isometric classes of \(D(K/{\mathbb{Q}})^{-1}\)-modular (skew) Hermitian forms over the ring \({\mathfrak O}_ K\) of integers of the cyclotomic field \(K:={\mathbb{Q}}(X)/(\Phi)\). This can be viewed as an integral version of the principle of Milnor and Springer-Steinberg. Applying it to the case of regular elements \((\phi (m)=2n)\) the author gets (Theorem 2): The number of conjugacy classes in \(\Gamma_ n(\Phi)\) is equal to \(h^-(K)\), the relative class number of K, times \(2^{n+t- 1}\) which is the number of integral conjugacy classes in Sp(n,\({\mathbb{Q}})(\Phi)\) of Sp(n,\({\mathbb{R}})(\Phi)\) \((t=1\) if K has prime power conductor, and \(t=0\) otherwise). This refines a result of \textit{H. Midorikawa} [Tokyo J. Math. 6, 25-38 (1983; Zbl 0522.10017)]. In Theorem 3, the characterization of the integral conjugacy classes among the \(2^ n\) classes in Sp(n,\({\mathbb{R}})(\Phi)\), by their eigenvalues as elements of U(n), the unitary group of degree n is given.
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 3912473
zbMATH Keywords: elliptic conjugacy classes
zbMATH Keywords: Siegel modular group
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: Hermitian forms
zbMATH Keywords: cyclotomic field
zbMATH Keywords: class number
zbMATH Keywords: integral conjugacy classes
MaRDI profile type: Publication
full work available at URL: pjaa.61.74
OpenAlex ID: W2065407828
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DOI: 10.3792/PJAA.61.74
description: scientific article; zbMATH DE number 3912473