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Séminaire de Théorie des Nombres, Paris 1985–86 pp 67–81Cite as

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Some Recent Results on Complex Powers and Zeta Distributions

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Part of the book series: Progress in Mathematics ((PM,volume 71))

Abstract

This is an expanded and updated version of our talk given on Dec. 2, 1985 in the Séminaire de Théorie des Nombres de Paris under the title, “Complex powers and their functional equations”. We have tried to state definitions precisely and results concisely. We have included some remarks not stated elsewhere. As we emphasized in the talk, a problem of Weil is the underlying theme.

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Bibliography

  1. M.F. Atiyah.- Resolution of singularities and division of distributions, Comm. pure and appl. math. 23 (1970), 145–150.

    Article  MathSciNet  MATH  Google Scholar 

  2. I.N. Bernshtein and S.I. Gel’fand.- Meromorphic property of the functions PA, Functional Analysis and its Applications 3 (1969), 68–69.

    Google Scholar 

  3. I.N. Bernshtein.- The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and its Applications 6 (1972), 273–285.

    Article  Google Scholar 

  4. B.A. Datskovsky.- On zeta functions associated with the space of binary cubic forms with coefficients in a function field, Thesis, Harvard, 1984.

    Google Scholar 

  5. J. Denef.- The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984), 1–23.

    MathSciNet  MATH  Google Scholar 

  6. J. Denef.- On the degree of Igusa’s local zeta function, preprint.

    Google Scholar 

  7. I.M. Gel’fand and G.E. Shilov.- Generalized functions. I, Academic Press (1964).

    Google Scholar 

  8. J. Igusa.- Complex powers and asymptotic expansions. I. Crelles J. Math. 268/269 (1974), 110–130; II. Ibid. 278–279 (1975), 307–321.

    Google Scholar 

  9. J. Igusa.- Some results on p-adic complex powers, Amer. J. Math. 106 (1984), 1013–1032.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Igusa.- On functional equations of complex powers, Invent. math. 85 (1986), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Igusa.- On a certain class of prehomogeneous vector spaces, to appear in J. of Algebra.

    Google Scholar 

  12. J. Igusa.- Zeta distributions associated with some invariants, to appear in Amer. J. Math.

    Google Scholar 

  13. V.G. Kac, V.L. Popov and E.B. Vinberg.- Sur les groupes linéaires algébriques dont l’algèbre des invariants est libres, C. R. Acad. Sc. Paris 283 (1976), 875–878.

    MathSciNet  MATH  Google Scholar 

  14. M. Kashiwara.- B-functions and holonomic systems (Rationality of b-functions), Invent. Math. 38 (1976), 33–53.

    Google Scholar 

  15. T. Kimura.- The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 (1982), 1–80.

    MATH  Google Scholar 

  16. T. Ono.- An integral attached to a hypersurface, Amer. J. Math. 90 (1968), 1224–1236.

    Article  MATH  Google Scholar 

  17. Kimura.- A classification of irreducible vector spaces and their relative invariants, 65 (1977), 1–155.

    Google Scholar 

  18. M. Sato and T.Shintani.- On zeta functions associated with vector spaces, Ann. Math. 100 (1974)

    Google Scholar 

  19. J.-P. Serre.- Cohomologie Galoisienne.- Lect. Notes in Math. 5, Springer-Verlag (1965).

    Google Scholar 

  20. J.-P. Serre.- Méthodes adéliques.- Résumé des cours et travaux, Collège de France (1981–1982), 81–89.

    Google Scholar 

  21. T. Shintani.- On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188.

    Article  MathSciNet  Google Scholar 

  22. J. Tate.- Fourier analysis in number fields and Hecke’s zeta-functions, Thesis, Princeton, 1950; Algebraic Number Theory, Acad. Press. (1967), 305–347.

    Google Scholar 

  23. A. Weil.- Adeles and algebraic groups.- Institute for Advanced Study, Princeton (1961); Progress in Math. 23, Birkhäuser (1982).

    Google Scholar 

  24. A. Weil.- Fonction zêta et distributions, Sém. Bourbaki 312 (1966), 1–9; Collected Papers III, Springer-Verlag (1980), 158–163.

    Google Scholar 

  25. D.J. Wright.- The adelic zeta function associated to the space of binary cubic forms, Math. Ann. 270 (1985), 503–534.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    Jun-Ichi Igusa

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  1. Jun-Ichi Igusa
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Editors and Affiliations

  1. Centre d’Orsay, Université de Paris-Sud, F-91405, Orsay Cedex, France

    Catherine Goldstein

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© 1987 Springer Science+Business Media New York

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Igusa, JI. (1987). Some Recent Results on Complex Powers and Zeta Distributions. In: Goldstein, C. (eds) Séminaire de Théorie des Nombres, Paris 1985–86. Progress in Mathematics, vol 71. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-4267-1_5

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  • DOI: https://doi.org/10.1007/978-1-4757-4267-1_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4757-4268-8

  • Online ISBN: 978-1-4757-4267-1

  • eBook Packages: Springer Book Archive

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