THE FUNDAMENTAL THEOREM OF PREHOMOGENEOUS VECTOR SPACES MODULO p(m) (WITH AN APPENDIX BY F. SATO)

摘要

For a number field K with ring of integers O-K, we prove an analogue over finite rings of the form O-K/P-m of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where P is a big enough prime ideal Of OK and m > 1. In the appendix, F. Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J. Denef and A. Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113 (1998), 237-346] to the functional equation of L-functions of Dirichlet type associated with prehomogeneous vector spaces.