The Fundamental Theorem of prehomogeneous vector spaces modulo $p^m$. With an appendix "L-functions of prehomogeneous vector spaces" by Fumihiro Sato

Abstract

For a number field $K$ with ring of integers ${\mathcal{O}}_K$, we prove an analogue over finite rings of the form ${\mathcal{O}}_K/{\mathcal{P}}^m$ of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $\mathcal{P}$ is a big enough prime ideal of ${\mathcal{O}}_K$ and $m>1$. In the appendix, F. Sato gives an application of the Theorems A, B and the Theorems A, B, C in J. Denef and A. Gyoja [\emph{Character sums associated to prehomogeneous vector spaces}, Compos. Math., \textbf{113} (1998) 237--346] to the functional equation of $L$-functions of Dirichlet type associated with prehomogeneous vector spaces.