The fundamental theorem of prehomogeneous vector spaces modulo p m (With an appendix by F. Sato)
[Théorème fondamental des espaces vectoriels préhomogènes modulo p m . Avec un appendice par F. Sato]
Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 4, pp. 475-494.

Soit K un corps de nombres avec anneaux d’entiers 𝒪 K  ; nous prouvons un analogue, sur des anneaux finis de la forme 𝒪 K /𝒫 m , du théorème fondamental sur la transformation de Fourier de l’invariante relative d’un espace vectoriel préhomogène. Ici, 𝒫 est un idéal premier assez grand de 𝒪 K et m>1. Dans l’appendice, F. Sato donne une application des théorèmes 1.1, 1.3 et des théorèmes A, B, C de J.Denef et A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113 (1998), 237-346] à l’équation fonctionelle de L-fonctions de type Dirichlet associées aux espaces vectorielles préhomogènes.

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.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.

DOI : 10.24033/bsmf.2543
Classification : 11S90, 11L07, 11M41, 11T24, 11L05, 20G40
Keywords: prehomogeneous vector spaces, $L$-functions, Bernstein-Sato polynomial, fundamental theorem of prehomogeneous vector spaces, exponential sums
Mot clés : espaces vectorielles préhomogènes, $L$-fonctions, polynôme de Bernstein-Sato, théorème fondamental des espaces vectorielles préhomogènes, sommes exponentielles
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Cluckers, Raf; Herremans, Adriaan. The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato). Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 4, pp. 475-494. doi : 10.24033/bsmf.2543. http://www.numdam.org/articles/10.24033/bsmf.2543/

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