Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms
type: Item
instance of: scholarly article
title: Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms
zbMATH Open document ID: 0585.10017
DOI: 10.1016/0022-314X(86)90007-7
author: Ki-ichiro Hashimoto
author: Robert J. Sibner
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: It is appropriate to extract some sentences from the authors' summary: We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n,\({\mathbb{Z}})\) which satisfy \(X^ 2+I=0\) \((resp.\quad X^ 2+X+I=0)\) and the set of Hermitian forms of rank n over \({\mathbb{Z}}[\sqrt{-1}]\) \((resp.\quad {\mathbb{Z}}[(-1+\sqrt{- 3})/2])\) of determinant \(\pm 1\). As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of \(m^ 2+1\) (resp. \(m^ 2+m+1)\) can be represented by the quadratic form \(F(X,Y)=X^ 2+Y^ 2\) (resp. \(X^ 2+XY+Y^ 2)\) with relatively prime integers X,Y.
reviewed by: Michio Ozeki
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 3937250
zbMATH Keywords: conjugacy classes of elliptic transformations
zbMATH Keywords: Hermitian forms of
zbMATH Keywords: rank n
zbMATH Keywords: positive symmetric integral matrices
zbMATH Keywords: mass formula for
zbMATH Keywords: Hermitian forms
MaRDI profile type: Publication
cites work: The conjugacy classes in the unitary, symplectic and orthogonal groups over an algebraic number field
cites work: Q3898575
cites work: Representations of positive definite quadratic forms.
cites work: Class numbers of definite Hermitian forms
cites work: A formula for the number of semi-simple conjugacy classes in the arithmetic subgroups
cites work: On certain elliptic conjugacy classes of the Siegel modular group
cites work: Q3737498
full work available at URL: 0022-314x(86)90007-7
OpenAlex ID: W2047269233
Recommended article: Siegel's modular forms and the arithmetic of quadratic forms
Recommended article: Q3979373
Recommended article: Siegel modular forms. A classical and representation-theoretic approach
Recommended article: Generalized transvectants and Siegel modular forms
Recommended article: SPECIAL QUADRATIC FORMS, SIEGEL MODULAR GROUPS AND SIEGEL MODULAR VARIETIES
Recommended article: Galois representations for holomorphic Siegel modular forms
Recommended article: On convolutions of Siegel modular forms
Recommended article: Q5380812
Recommended article: Q4665014
Recommended article: Siegel Modular Forms and Their Applications
description: scientific article; zbMATH DE number 3937250