The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)

With an Appendix by F. Sato

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 O K and m > 1. In the appendix, F. to the functional equation of L-functions of Dirichlet type associated with prehomogeneous vector spaces.

COMPUTATION OF CHARACTER SUMS AND APPLICATIONS TO PREHOMOGENEOUS VECTOR SPACES 1 with an appendix ” L - FUNCTIONS OF PREHOMOGENEOUS VECTOR SPACES

For an arbitrary number field K with ring of integers OK , we prove an analogue over finite rings of the form OK/p 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 1.2 and 1.5 (and Theorems A, B, C in [4]) to the functiona...

Prehomogeneous Vector Spaces and Field Extensions Ii

Free Divisors in Prehomogeneous Vector Spaces

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coinci...

Solution Spaces of H-Systems and the Ore-Sato Theorem∗

An H-system is a system of first-order linear homogeneous difference equations for a single unknown function T , with coefficients which are polynomials with complex coefficients. We consider solutions of H-systems which are of the form T : dom(T )→ C where either dom(T ) = Z, or dom(T ) = Z \ S and S is the set of integer singularities of the system. It is shown that any natural number is the ...