The fundamental theorem of prehomogeneous vector spaces modulo p m (With an appendix by F. Sato)
Bulletin de la Société Mathématique de France, Volume 135 (2007) no. 4, p. 475-494

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.

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.

DOI : https://doi.org/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
@article{BSMF_2007__135_4_475_0,
     author = {Cluckers, Raf and Herremans, Adriaan},
     title = {The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {135},
     number = {4},
     year = {2007},
     pages = {475-494},
     doi = {10.24033/bsmf.2543},
     zbl = {1207.11118},
     mrnumber = {2439196},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2007__135_4_475_0}
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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, Volume 135 (2007) no. 4, pp. 475-494. doi : 10.24033/bsmf.2543. http://www.numdam.org/item/BSMF_2007__135_4_475_0/

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