On a classification of prehomogeneous vector spaces over local and global fields
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.
\(F\)-forms of prehomogeneous vector spaces
scientific article; zbMATH DE number 1014022
scientific article