Gauss sums of prehomogeneous vector spaces
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.
Gauss sums of prehomogeneous vector space
prehomogeneous vector spaces defined over finite fields
scientific article; zbMATH DE number 3900945