Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T23:54:45.136Z Has data issue: false hasContentIssue false
  • English
  • Français

Motivic zeta functions for prehomogeneous vector spaces and castling transformations

Published online by Cambridge University Press:  22 January 2016

François Loeser
François Loeser*
Affiliation:
École Normale Supérieure, Département de mathématiques et applications, 45 rue d’Ulm, 75230 Paris Cedex 05, France, (UMR 8553 du CNRS), [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the behaviour of motivic zeta functions of prehomogeneous vector spaces under castling transformations. In particular we deduce how the motivic Milnor fibre and the Hodge spectrum at the origin behave under such transformations.

Keywords

11S9014L3014L24
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 171 , 2003 , pp. 85 - 105
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[1] Denef, J. and Loeser, F., Motivic Igusa zeta functions, J. Algebraic Geom., 7 (1998), 505–537.Google Scholar
[2] Denef, J. and Loeser, F., Geometry on arc spaces of algebraic varieties, Proceedings of 3rd European Congress of Mathematics, Barcelona 2000, Progress in Mathematics 201, Birkhaüser (2001), pp. 327–348.Google Scholar
[3] Denef, J. and Loeser, F., Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology, 41 (2002), 1031–1040.Google Scholar
[4] Gordon, J., Motivic Haar measure on reductive groups, preprint, March 2002, available at math.AG/0203106.Google Scholar
[5] Guibert, G., Fonction zêta motivique associée à une famille de séries de deux variables, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 457–460.Google Scholar
[6] Hosokawa, H., Igusa local zeta functions and parabolic castling transformation of prehomogeneous vector spaces, J. Number Theory, 74 (1999), 148–171.CrossRefGoogle Scholar
[7] Igusa, J., On functional equations of complex powers, Invent. Math., 85 (1986), 1–29.Google Scholar
[8] Igusa, J., On the arithmetic of a singular invariant, Amer. J. Math., 110 (1988), 197–233.Google Scholar
[9] Igusa, J., b-functions and p-adic integrals, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 231–241.Google Scholar
[10] Igusa, J., Local zeta functions of certain prehomogeneous vector spaces, Amer. J. Math., 114 (1992), 251–296.Google Scholar
[11] Kimura, T., The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J., 85 (1982), 1–80.Google Scholar
[12] Libgober, A., Hodge decomposition of Alexander invariants, preprint, October 2001, available at math.AG/0108018.Google Scholar
[13] Looijenga, E., Motivic Measures, Astérisque 276, Séminaire Bourbaki, exposé 874 (2002), 267–297.Google Scholar
[14] Sabbah, C., Modules d’Alexander et D-modules, Duke Math. J., 60 (1990), 729–814.CrossRefGoogle Scholar
[15] Saito, M., Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24 (1988), 849–995.Google Scholar
[16] Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26 (1990), 221–333.Google Scholar
[17] Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1–155.Google Scholar
[18] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces, Ann. of Math., 100 (1974), 131–170.Google Scholar
[19] Teranishi, Y., Relative invariants and b-functions of prehomogeneous vector spaces , Nagoya Math. J., 98 (1985), 139–156.CrossRefGoogle Scholar
[20] Steenbrink, J., Mixed Hodge structures on the vanishing cohomology, Real and Complex Singularities, Sijthoff and Noordhoff, Alphen aan den Rijn (1977), 525–563.Google Scholar
[21] Steenbrink, J., The spectrum of hypersurface singularities, Théorie de Hodge, Luminy 1987, Astérisque, 179–180 (1989), 163–184.Google Scholar
[22] Varchenko, A., Asymptotic Hodge structure in the vanishing cohomology, Math. USSR Izvestija, 18 (1982), 469–512.Google Scholar
[23] Yano, T. and Ozeki, I., The b-function of a prehomogeneous vector space (SL(5) × GL(4), Λ2 ⊗ Λ1). Microlocal structure of the regular prehomogeneous vector space associated with SL(5) × GL(4). II, Research on prehomogeneous vector spaces (Kyoto, 1996), RIMS Kōkyūroku, 999 (1997), 92–115.Google Scholar