Arithmetic calculus of Fourier transforms by Igusa local zeta functions

Author:
Tatsuo Kimura

Journal:
Trans. Amer. Math. Soc. **346** (1994), 297-306

MSC:
Primary 11S40; Secondary 11S80, 20G20

DOI:
https://doi.org/10.1090/S0002-9947-1994-1267223-5

MathSciNet review:
1267223

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show the possibility of explicit calculation of the Fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over by using the explicit form of -adic Igusa local zeta functions.

**[Gyoja]**A. Gyoja,*Functional equations for Igusa local zeta functions*, in preparation.**[Hosokawa]**Hiroshi Hosokawa,*Some results on Igusa local zeta functions associated with simple prehomogeneous vector spaces*, J. Math. Soc. Japan**49**(1997), no. 3, 565–587. MR**1452703**, https://doi.org/10.2969/jmsj/04930565**[Igusa 1]**Jun-ichi Igusa,*Some results on 𝑝-adic complex powers*, Amer. J. Math.**106**(1984), no. 5, 1013–1032. MR**761577**, https://doi.org/10.2307/2374271**[Igusa 2]**Jun-ichi Igusa,*On functional equations of complex powers*, Invent. Math.**85**(1986), no. 1, 1–29. MR**842045**, https://doi.org/10.1007/BF01388789**[Igusa 3]**Jun-ichi Igusa,*On a certain class of prehomogeneous vector spaces*, J. Pure Appl. Algebra**47**(1987), no. 3, 265–282. MR**910424**, https://doi.org/10.1016/0022-4049(87)90051-X**[Igusa 4]**Jun-ichi Igusa,*Zeta distributions associated with some invariants*, Amer. J. Math.**109**(1987), no. 1, 1–33. MR**878195**, https://doi.org/10.2307/2374548**[Iwasawa]**K. Iwasawa,*A note on*-*functions*, Proc. Internat. Congress Math. (Cambridge, Mass., 1950), vol. 1, Amer. Math. Soc., Providence, RI, 1952, p. 322;*A letter to J. Dieudonné*(1952), Appendix in Adv. Stud. Pure Math.**21**(1992).**[K-K]**Tatsuo Kimura and Takeyoshi Kogiso,*On adelic zeta functions of prehomogeneous vector spaces with finitely many adelic open orbits*, Zeta functions in geometry (Tokyo, 1990) Adv. Stud. Pure Math., vol. 21, Kinokuniya, Tokyo, 1992, pp. 21–31. MR**1210781**, https://doi.org/10.2969/aspm/02110021**[K-K-H]**T. Kimura, S. Kasai, and H. Hosokawa,*Universal transitivity of simple and 2-simple prehomogeneous vector spaces*, Ann. Inst. Fourier (Grenoble)**38**(1988), no. 2, 11–41 (English, with French summary). MR**949009****[K-K-M]**M. Kashiwara, T. Kimura, and M. Muro,*Microlocal calculus of Fourier transforms over*, in preparation.**[Kimura]**Tatsuo Kimura,*The 𝑏-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces*, Nagoya Math. J.**85**(1982), 1–80. MR**648417****[M]**Masakazu Muro,*Microlocal analysis and calculations on some relatively invariant hyperfunctions related to zeta functions associated with the vector spaces of quadratic forms*, Publ. Res. Inst. Math. Sci.**22**(1986), no. 3, 395–463. MR**861776**, https://doi.org/10.2977/prims/1195177844**[Ono]**Takashi Ono,*An integral attached to a hypersurface*, Amer. J. Math.**90**(1968), 1224–1236. MR**238851**, https://doi.org/10.2307/2373298**[F. Sato 1]**Fumihiro Sat\B{o},*Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations*, Tohoku Math. J. (2)**34**(1982), no. 3, 437–483. MR**676121**, https://doi.org/10.2748/tmj/1178229205**[F. Sato 2]**Fumihiro Sat\B{o},*Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations*, Tohoku Math. J. (2)**34**(1982), no. 3, 437–483. MR**676121**, https://doi.org/10.2748/tmj/1178229205**[S-K]**M. Sato and T. Kimura,*A classification of irreducible prehomogeneous vector spaces and their relative invariants*, Nagoya Math. J.**65**(1977), 1–155. MR**430336****[S-S]**Mikio Sato and Takuro Shintani,*On zeta functions associated with prehomogeneous vector spaces*, Ann. of Math. (2)**100**(1974), 131–170. MR**344230**, https://doi.org/10.2307/1970844**[Tate]**J. T. Tate,*Fourier analysis in number fields, and Hecke’s zeta-functions*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR**0217026**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11S40,
11S80,
20G20

Retrieve articles in all journals with MSC: 11S40, 11S80, 20G20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1267223-5

Keywords:
Prehomogeneous vector spaces,
Iwasawa-Tate theory

Article copyright:
© Copyright 1994
American Mathematical Society