Gauss sums of prehomogeneous vector spaces
type: Item
instance of: scholarly article
title: Gauss sums of prehomogeneous vector spaces
zbMATH Open document ID: 0565.14032
author: Akihiko Gyoja
author: Noriaki Kawanaka
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: This paper is an announcement of the authors' results on prehomogeneous vector spaces defined over finite fields. Let (G,V) be a regular irreducible prehomogeneous vector space. We put f(x) an irreducible relative invariant. Then, we get an irreducible regular prehomogeneous vector space defined over a finite field \({\mathbb{F}}_ p\) with a prime number p by the reduction modulo p because (G,V) is defined over \({\mathbb{Z}}\). Let \({\mathbb{F}}_ q\) be a finite extension of \({\mathbb{F}}_ p\), \(\psi\) : \({\mathbb{F}}_ q\to {\mathbb{C}}^{\times}\) a nontrivial additive character and \(\chi\) : \({\mathbb{F}}_ q\to {\mathbb{C}}^{\times}\) a multiplicative character. We put \(\chi (0)=0\). The order of \(\chi \in Hom({\mathbb{F}}_ q,{\mathbb{C}}^{\times})\) is denoted by ord(\(\chi)\). The authors define: \({\mathcal F}_{\psi}[\chi (f)](v^{\vee})=\sum_{v\in V({\mathbb{F}}_ q)}\chi (f(v))\psi (<v,v^{\vee}>),\) and call it the ''Gauss sum'' of (G,V). Here \(V({\mathbb{F}}_ q)\) is the set of \({\mathbb{F}}_ q\)-points of V and \(v^{\vee}\) is an element of the dual space \(V^{\vee}({\mathbb{F}}_ q)\). They have got a conjecture on the explicit calculation of \({\mathcal F}_{\psi}[\chi (f))\) and prove it in almost all cases by using micro-local analysis.
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 32A45
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 3900945
zbMATH Keywords: Gauss sums of prehomogeneous vector space
zbMATH Keywords: prehomogeneous vector spaces defined over finite fields
zbMATH Keywords: micro-local analysis
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: pjaa.61.19
OpenAlex ID: W2038518010
cites work: Q3346429
cites work: Q3320507
cites work: La conjecture de Weil. II
cites work: The invariant holonomic system on a semisimple Lie algebra
cites work: B-functions and holonomic systems. Rationality of roots of B-functions
cites work: The Riemann-Hilbert problem for holonomic systems
cites work: Transformation de Fourier et majoration de sommes exponentielles. (Fourier transformation and majoration of exponential sums.)
cites work: Theb-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces
cites work: Micro-local analysis of prehomogeneous vector spaces
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: On zeta functions associated with prehomogeneous vector spaces
DOI: 10.3792/PJAA.61.19
description: scientific article; zbMATH DE number 3900945