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

Cluckers, Raf; Herremans, Adriaan

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.

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

[1] Z. I. Borevitch & I. R. Chafarevitch - Théorie des nombres, Traduit par Myriam et Jean-Luc Verley. Traduction faite d'après l'édition originale russe. Monographies Internationales de Mathématiques Modernes, No. 8, Gauthier-Villars, 1967. | MR | Zbl

[2] N. Bourbaki - Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles, No. 1333, Hermann, 1967. | MR | Zbl

[3] R. Cluckers & A. Herremans - « The Fundamental Theorem of prehomogeneous vector spaces modulo $p^m$ », main body of this article. | Numdam | Zbl

[4] B. Datskovsky & D. J. Wright - « The adelic zeta function associated to the space of binary cubic forms. II. Local theory », J. reine angew. Math. 367 (1986), p. 27-75. | EuDML | MR | Zbl

[5] J. Denef & A. Gyoja - « Character sums associated to prehomogeneous vector spaces », Compositio Math. 113 (1998), p. 273-346. | MR | Zbl

[6] A. Gyoja - « Theory of prehomogeneous vector spaces without regularity condition », Publ. Res. Inst. Math. Sci. 27 (1991), p. 861-922. | MR | Zbl

[7] J 106 (1984), p. 1013-1032. | MR | Zbl

[8] -, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, 2000. | MR | Zbl

[9] D. Kazhdan & A. Polishchuk - « Generalized character sums associated to regular prehomogeneous vector spaces », Geom. Funct. Anal. 10 (2000), p. 1487-1506. | MR | Zbl

[10] J. Milnor - Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, vol. 51, Princeton University Press, 1963. | MR | Zbl

[11] H. Saito - « A generalization of Gauss sums and its applications to Siegel modular forms and $L$-functions associated with the vector space of quadratic forms », J. reine angew. Math. 416 (1991), p. 91-142. | MR | Zbl

[12] -, « On $L$-functions associated with the vector space of binary quadratic forms », Nagoya Math. J. 130 (1993), p. 149-176. | MR | Zbl

[13] -, « Convergence of the zeta functions of prehomogeneous vector spaces », Nagoya Math. J. 170 (2003), p. 1-31. | MR | Zbl

[14] F. Sato - « L-functions of prehomogeneous vector spaces », Appendix of this article.

[15] -, « Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations », Tōhoku Math. J. (2) 34 (1982), p. 437-483. | MR | Zbl

[16] -, « On functional equations of zeta distributions », Adv. Stud. Pure Math. 15 (1989), p. 465-508. | MR | Zbl

[17] M. Sato - « Theory of prehomogeneous vector spaces », Sugaku no Ayumi 15 (1970), p. 85-157, notes by T.Shintani. | Zbl

[18] -, « Theory of prehomogeneous vector spaces (algebraic part)-the English translation of Sato's lecture from Shintani's note », Nagoya Math. J. 120 (1990), p. 1-34. | MR | Zbl

[19] M. Sato & T. Kimura - « A classification of irreducible prehomogeneous vector spaces and their relative invariants », Nagoya Math. J. 65 (1977), p. 1-155. | MR | Zbl

[20] J-P. Serre - « Quelques applications du théorème de densité de Chebotarev », Publ. Math. I.H.E.S. 54 (1981), p. 323-401. | Numdam | MR | Zbl

[21] H. M. Stark - « $L$-functions and character sums for quadratic forms. I », Acta Arith. 14 (1967/1968), p. 35-50. | MR | Zbl

[22] T. Ueno - « Elliptic modular forms arising from zeta functions in two variables attached to the space of binary Hermitian forms », J. Number Theory 86 (2001), p. 302-329. | MR | Zbl

[23] -, « Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms », 2004, to appear in Nagoya Math. J., p. 1-37. | MR

[24] A. Weil - Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer New York, Inc., New York, 1967. | MR | Zbl