Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T13:43:16.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Local zeta functions and Newton polyhedra

Published online by Cambridge University Press:  22 January 2016

W. A. Zuniga-Galindo
W. A. Zuniga-Galindo*
Affiliation:
Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue Miami Shores, Florida 33161, [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.

To a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

Keywords

11S4011D7911L05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 172 , 2003 , pp. 31 - 58
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[A] Artin, M., Algebraic approximations of structures over complete local rings, Pub. I.H.E.S., 36, 23–58.Google Scholar
[D1] Denef, J., The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math., 77 (1984), 1–23.CrossRefGoogle Scholar
[D2] Denef, J., Report on Igusa’s local zeta functions, Seminaire Bourbaki 741, Astérisque, 201/202/203 (1990–1991).Google Scholar
[D3] Denef, J., Poles of p-adic complex powers and Newton Polyhedra, Nieuw Archief voor Wiskunde, 13 (1995), 289–295.Google Scholar
[D-H] Denef, J. and Hoornaert, K., Newton polyhedra and Igusa’s local zeta functions, Journal of Number Theory, 89 (2001), 31–64.CrossRefGoogle Scholar
[D-S-1] Denef, J. and Sargos, P., Polyédre de Newton et distribution J. Analyse Math., 53 (1989), 201–218.CrossRefGoogle Scholar
[D-S-2] Denef, J. and Sargos, P., Polyédre de Newton et distribution . II, Math. Ann., 293 (1992), 193–211.CrossRefGoogle Scholar
[D-Sp] Denef, J. and Sperber, S., On exponential sums mod pm and Newton polyhedra, Bull. Belg. Math. Soc. Simon Stevin (2001), 55–63.Google Scholar
[I1] Igusa, J.-I., On the first terms of certain asymptotic expansions, Complex and Algebraic Geometry, Iwanami Shoten and Cambridge university press (1977), pp. 357–368.CrossRefGoogle Scholar
[I2] Igusa, J.-I., Complex powers and asymptotic expansions I, J. Reine Angew Math., 268/269 (1974), 110–130.Google Scholar
[I3] Igusa, J.-I., A stationary phase formula for p-adic integrals and its applications, Algebraic geometry and its applications, Springer-Verlag (1994), pp. 175–194.CrossRefGoogle Scholar
[I4] Igusa, J.-I., On the arithmetic of a singular invariant, Amer. J. Math. (1988), 197–233.CrossRefGoogle Scholar
[K-M-S] Kempf, G., Knudsend, F., Mumford, D. and Saint-Donat, B., Toroidal embedings, Lectures notes in Mathematics, 339 (1973).CrossRefGoogle Scholar
[L-M] Lichtin, B. and Meuser, D., Poles of a local zeta function and Newton polygons, Compositio Math., 55 (1985), 313–332.Google Scholar
[Me] Meuser, D., On the poles of a local zeta function for curves, Invent. Math., 73 (1983), 445–465.CrossRefGoogle Scholar
[N] Néron, A., Modéles minimaux des variétes abéliennes sur corps locaux et globaux, Pub. Math. I.H.E.S., 21 (1964).Google Scholar
[Va] Varchenko, A., Newton polyhedra and estimation of oscillatory integrals, Functional Anal. Appl., 10 (1977), 175–196.CrossRefGoogle Scholar
[Ve] Veys, W., On the poles of Igusa’s local zeta functions for curves, J. London Math. Soc., 41 (1990), 27–32.CrossRefGoogle Scholar
[Z-G] Zuniga-Galindo, W. A., Igusa’s local zeta functions of semiquasihomogeneous polynomials, Trans. Amer. Math. Soc., 353 (2001), 3193–3207.CrossRefGoogle Scholar