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

Raf Cluckers; Adriaan Herremans

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 4, page 475-494
  • ISSN: 0037-9484

Abstract

top
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.

How to cite

top

Cluckers, Raf, and 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 135.4 (2007): 475-494. <http://eudml.org/doc/272426>.

@article{Cluckers2007,
abstract = {For a number field $K$ with ring of integers $\{\mathcal \{O\}\}_K$, we prove an analogue over finite rings of the form $\{\mathcal \{O\}\}_K/\{\mathcal \{P\}\}^m$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $\{\mathcal \{P\}\}$ is a big enough prime ideal of $\{\mathcal \{O\}\}_K$ and $m&gt;1$. 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.},
author = {Cluckers, Raf, Herremans, Adriaan},
journal = {Bulletin de la Société Mathématique de France},
keywords = {prehomogeneous vector spaces; $L$-functions; Bernstein-Sato polynomial; fundamental theorem of prehomogeneous vector spaces; exponential sums},
language = {eng},
number = {4},
pages = {475-494},
publisher = {Société mathématique de France},
title = {The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)},
url = {http://eudml.org/doc/272426},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Cluckers, Raf
AU - Herremans, Adriaan
TI - The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 4
SP - 475
EP - 494
AB - For a number field $K$ with ring of integers ${\mathcal {O}}_K$, we prove an analogue over finite rings of the form ${\mathcal {O}}_K/{\mathcal {P}}^m$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where ${\mathcal {P}}$ is a big enough prime ideal of ${\mathcal {O}}_K$ and $m&gt;1$. 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.
LA - eng
KW - prehomogeneous vector spaces; $L$-functions; Bernstein-Sato polynomial; fundamental theorem of prehomogeneous vector spaces; exponential sums
UR - http://eudml.org/doc/272426
ER -

References

top
  1. [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. Zbl0145.04901MR205908
  2. [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. Zbl0206.50402MR219078
  3. [3] R. Cluckers & A. Herremans – « The Fundamental Theorem of prehomogeneous vector spaces modulo p m », main body of this article. Zbl1207.11118
  4. [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. Zbl0575.10016MR839123
  5. [5] J. Denef & A. Gyoja – « Character sums associated to prehomogeneous vector spaces », Compositio Math.113 (1998), p. 273–346. Zbl0919.11086MR1644996
  6. [6] A. Gyoja – « Theory of prehomogeneous vector spaces without regularity condition », Publ. Res. Inst. Math. Sci.27 (1991), p. 861–922. Zbl0773.14025MR1145669
  7. [7] J.-i. Igusa – « Some results on p -adic complex powers », Amer. J. Math. 106 (1984), p. 1013–1032. Zbl0589.14023MR761577
  8. [8] —, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, 2000. Zbl0959.11047MR1743467
  9. [9] D. Kazhdan & A. Polishchuk – « Generalized character sums associated to regular prehomogeneous vector spaces », Geom. Funct. Anal.10 (2000), p. 1487–1506. Zbl1001.11053MR1810750
  10. [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. Zbl0108.10401MR163331
  11. [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. Zbl0717.11053MR1099947
  12. [12] —, « On L -functions associated with the vector space of binary quadratic forms », Nagoya Math. J.130 (1993), p. 149–176. Zbl0774.11052MR1223734
  13. [13] —, « Convergence of the zeta functions of prehomogeneous vector spaces », Nagoya Math. J.170 (2003), p. 1–31. Zbl1045.11083MR1994885
  14. [14] F. Sato – « L-functions of prehomogeneous vector spaces », Appendix of this article. 
  15. [15] —, « Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations », Tōhoku Math. J. (2) 34 (1982), p. 437–483. Zbl0497.14007MR676121
  16. [16] —, « On functional equations of zeta distributions », Adv. Stud. Pure Math.15 (1989), p. 465–508. Zbl0714.11053MR1040618
  17. [17] M. Sato – « Theory of prehomogeneous vector spaces », Sugaku no Ayumi 15 (1970), p. 85–157, notes by T.Shintani. Zbl0715.22014
  18. [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. Zbl0715.22014MR1086566
  19. [19] M. Sato & T. Kimura – « A classification of irreducible prehomogeneous vector spaces and their relative invariants », Nagoya Math. J.65 (1977), p. 1–155. Zbl0321.14030MR430336
  20. [20] J-P. Serre – « Quelques applications du théorème de densité de Chebotarev », Publ. Math. I.H.E.S.54 (1981), p. 323–401. Zbl0496.12011MR644559
  21. [21] H. M. Stark – « L -functions and character sums for quadratic forms. I », Acta Arith. 14 (1967/1968), p. 35–50. Zbl0198.37801MR227122
  22. [22] T. Ueno – « Elliptic modular forms arising from zeta functions in two variables attached to the space of binary Hermitian forms », J. Number Theory86 (2001), p. 302–329. Zbl1014.11031MR1813115
  23. [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. MR2085308
  24. [24] A. Weil – Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer New York, Inc., New York, 1967. Zbl0176.33601MR234930

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.