On a classification of prehomogeneous vector spaces over local and global fields
type: Item
instance of: scholarly article
title: On a classification of prehomogeneous vector spaces over local and global fields
zbMATH Open document ID: 0874.14046
author: Hiroshi Saito
published in: Journal of Algebra
publication date: 1997-11-11T00:00:00Z
review text: We say that \((G,\rho,V)\) where \(G\) is an algebraic group over a field \(F\) and \(\rho\) is a rational representation of \(G\) on the finite-dimensional \(F\)-vector space \(V\), is a prehomogeneous vector space over \(F\) if \(V_{\overline F}:= \overline F\otimes_FV\) has a Zariski open \(G_{\overline F}:=\overline F\otimes_FG\)-orbit, where \(\overline F\) is an algebraic closure of \(F\). It is called an \(F\)-form of a prehomogeneous vector space. The determination of all \(F\)-forms of prehomogeneous vector spaces is a fundamental problem, especially for the study of arithmetic properties like zeta functions of prehomogeneous spaces. The purpose of this paper is to classify forms of reductive, reduced, irreducible and regular prehomogeneous vector spaces over non-archimedean local fields of characteristic 0 and algebraic number fields.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1014022
zbMATH Keywords: zeta-functions
zbMATH Keywords: \(F\)-forms of prehomogeneous vector spaces
zbMATH Keywords: local fields
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: jabr.1996.6807
OpenAlex ID: W2093783804
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DOI: 10.1006/JABR.1996.6807
description: scientific article; zbMATH DE number 1014022