On the holomorphy conjecture for Igusa’s local zeta function
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- by Jan Denef and Willem Veys PDF
- Proc. Amer. Math. Soc. 123 (1995), 2981-2988 Request permission
Abstract:
To a polynomial f over a p-adic field K and a character $\chi$ of the group of units of the valuation ring of K one associates Igusa’s local zeta function $Z(s,f,\chi )$, which is a meromorphic function on $\mathbb {C}$. Several theorems and conjectures relate the poles of $Z(s,f,\chi )$ to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of $\chi$ does not divide the order of any eigenvalue of monodromy of f, then $Z(s,f,\chi )$ is holomorphic on $\mathbb {C}$. We prove mainly that if the holomorphy conjecture is true for $f({x_1}, \ldots ,{x_{n - 1}})$, then it is true for $f({x_1}, \ldots ,{x_{n - 1}}) + x_n^k$ with $k \geq 3$, and we give some applications.References
- J. Denef, Local zeta functions and Euler characteristics, Duke Math. J. 63 (1991), no. 3, 713–721. MR 1121152, DOI 10.1215/S0012-7094-91-06330-1 —, Report on Igusa’s local zeta function, Sém. Bourbaki 741, Astérisque 201/202/203 (1991), 359-386.
- J. Denef, Degree of local zeta functions and monodromy, Compositio Math. 89 (1993), no. 2, 207–216. MR 1255694
- Jan Denef and Diane Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), no. 6, 1135–1152. MR 1137535, DOI 10.2307/2374901 J. Igusa, Complex powers and asymptotic expansions I, J. Reine Angew. Math. 268/269 (1974), 110-130; II, 278/279 (1975), 307-321.
- Jun-ichi Igusa, Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546292
- Jun-ichi Igusa, Some aspects of the arithmetic theory of polynomials, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 20–47. MR 900822, DOI 10.1007/978-1-4899-6664-3_{2}
- Jun-ichi Igusa, $b$-functions and $p$-adic integrals, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 231–241. MR 992457
- Tatsuo Kimura, Fumihiro Sat\B{o}, and Xiao-Wei Zhu, On the poles of $p$-adic complex powers and the $b$-functions of prehomogeneous vector spaces, Amer. J. Math. 112 (1990), no. 3, 423–437. MR 1055652, DOI 10.2307/2374750
- F. Loeser, Fonctions d’Igusa $p$-adiques et polynômes de Bernstein, Amer. J. Math. 110 (1988), no. 1, 1–21 (French). MR 926736, DOI 10.2307/2374537
- François Loeser, Fonctions d’Igusa $p$-adiques, polynômes de Bernstein, et polyèdres de Newton, J. Reine Angew. Math. 412 (1990), 75–96 (French). MR 1079002, DOI 10.1515/crll.1990.412.75
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- Koichi Sakamoto, Milnor fiberings and their characteristic maps, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 145–150. MR 0372244
- M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), 90–96 (French). MR 293122, DOI 10.1007/BF01390095 A. Grothendieck, P. Deligne, and N. Katz, Groupes de monodromie en géométrie algébrique I, Lecture Notes in Math., vol. 288, Springer, New York, 1972.
- Willem Veys, Holomorphy of local zeta functions for curves, Math. Ann. 295 (1993), no. 4, 635–641. MR 1214952, DOI 10.1007/BF01444907
- Willem Veys, Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121 (1993), no. 4, 545–598 (English, with English and French summaries). MR 1254752, DOI 10.24033/bsmf.2219
- Willem Veys, On Euler characteristics associated to exceptional divisors, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3287–3300. MR 1308026, DOI 10.1090/S0002-9947-1995-1308026-3
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2981-2988
- MSC: Primary 11S40; Secondary 32S40
- DOI: https://doi.org/10.1090/S0002-9939-1995-1283546-4
- MathSciNet review: 1283546