## Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants. (Igusa’s local zeta functions in several variables, fiber integration and discriminants).(French)Zbl 0718.11061

This paper deals with some properties of the location of poles of the so- called Igusa’s zeta function. Let K be a field of characteristic 0. Consider the k-tuple polynomials $$F(x):=(f_ 1(x),...,f_ k(x))$$ on $$x=(x_ 1,...,x_ n)\in K^ n$$ and regard it as a morphism from $$K^ n$$ to $$K^ k$$. We suppose that each $$f_ i$$ vanishes at the origin. We define the integral $Z_ F(s_ 1,...,s_ k):=\int_{K^ n}| f_ 1(x)|_ K^{s_ 1}... | f_ k(x)|_ K^{s_ k} \phi (x) | dx|.$ Here, $$\phi$$ (x) is a test function with compact support; $$s=(s_ 1,...,s_ k)\in {\mathbb{C}}^ k$$; $$| |$$ is the normalized valuation; $$| dx|$$ is the Haar measure on $$K^ n$$. $$Z_ F$$ is convergent if the real parts of $$s_ i$$ are all sufficiently large, and it can be continued to the whole set $${\mathbb{C}}^ k$$ as a meromorphic function. We call it Igusa’s zeta function if K is a p-adic field. Thus it has possible poles whose locations makes a set of hyperplanes in $${\mathbb{C}}^ k$$. The set of the directions of the hyperplanes, we denote it by $${\mathcal P}(F)$$, is called “pents” of the morphism F. The author investigates the relation between $${\mathcal P}(F)$$ and $${\mathcal P}(\Delta_ F)$$, where $$\Delta_ F$$ is the discriminant of F. The principal result is that $${\mathcal P}(F)$$ is contained in $${\mathcal P}(\Delta_ F)$$ under the suitable condition that the author called “bon”.

### MSC:

 11S40 Zeta functions and $$L$$-functions 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

### Keywords:

location of poles; Igusa’s zeta function; p-adic field
Full Text:

### References:

 [1] M. F. ATIYAH , Resolution of Singularities and Division of Distributions (Comm. pure and appl. Math., vol. 23, 1970 , p. 145-150). MR 41 #815 | Zbl 0188.19405 · Zbl 0188.19405 [2] D. BARLET , Fonctions de type trace (Ann. Inst. Fourier, vol. 33, n^\circ 2, 1983 , p. 43-76). Numdam | MR 85c:32022 | Zbl 0498.32002 · Zbl 0498.32002 [3] D. BARLET et H.-M. MAIRE , Développements asymptotiques, transformation de Mellin complexe et intégration dans les fibres (Séminaire P. Lelong, 1986 ). Zbl 0649.32008 · Zbl 0649.32008 [4] I. N. BERNSTEIN et S. I. GELFAND , Meromorphic Property of the Functions P\lambda (Functional Analysis and its Applications, 3, 1969 , p. 68-69). MR 40 #723 | Zbl 0208.15201 · Zbl 0208.15201 [5] J. DENEF , Poles of p-Adic Complex Powers and Newton Polyhedra , Preprint, 1985 . MR 87i:11175 · Zbl 0591.14016 [6] J.-P. G. HENRY , M. MERLE et C. SABBAH , Sur la condition de Thom stricte pour un morphisme analytique complexe (Ann. Scient. Ec. Norm. Sup., 4e série, 17, 1984 , p. 227-268). Numdam | MR 86m:32019 | Zbl 0551.32012 · Zbl 0551.32012 [7] J.-I. IGUSA , Complex Powers and Asymptotic Expansions I, Functions of Certain Types (J. reine angew. Math., vol. 268/269, 1974 , p. 110-130) ; II, Asymptotic expansions (J. reine angew. Math., vol. 278/279, 1975 , p. 307-321). Article | MR 50 #254 | Zbl 0287.43007 · Zbl 0287.43007 [8] J.-I. IGUSA , Lectures on Forms of Higher Degree , Springer Verlag, 1978 . MR 80m:10020 | Zbl 0417.10015 · Zbl 0417.10015 [9] M. KASHIWARA et P. SCHAPIRA , Microlocal Study of Sheaves (Astérisque, n^\circ 128, Société Mathématique de France, 1985 ). MR 87f:58159 | Zbl 0589.32019 · Zbl 0589.32019 [10] LÊ D. T. , The Geometry of the Monodromy Theorem , Volume dédié à C.P. Ramanujam, Springer Verlag, 1978 . · Zbl 0434.32010 [11] LÊ D. T. , F. MICHEL et C. WEBER , Courbes polaires et topologie des courbes planes , Preprint. · Zbl 0748.32018 [12] F. LOESER , Fonctions d’Igusa p-adiques et polynômes de Bernstein (American Journal of Mathematics, vol. 110, 1988 , p. 1-21). MR 89d:11110 | Zbl 0644.12007 · Zbl 0644.12007 [13] F. LOESER , Fonctions |f|s, théorie de Hodge et polynômes de Bernstein-Sato. Géométrie algébrique et applications , Travaux en cours 24, Hermann, 1987 , p. 21-33. MR 89d:32017 | Zbl 0621.32018 · Zbl 0621.32018 [14] F. LOESER , Quelques résultats récents concernant les fonctions d’Igusa (Séminaire de Théorie des Nombres de Bordeaux, 1986 - 1987 , Exposé n^\circ 25). Article | Zbl 0644.12006 · Zbl 0644.12006 [15] C. SABBAH , Proximité évanescente I, la structure polaire d’un D-module, Appendice en collaboration avec F. Castro (Compositio Mathematica, vol. 62, 1987 , p. 283-328) ; II, Equations fonctionnelles pour plusieurs fonctions analytiques (Compositio Mathematica, vol. 64, 1987 , pp. 213-241). Numdam | Zbl 0632.32006 · Zbl 0632.32006 [16] C. SABBAH , Appendice à Proximité évanescente II , Manuscrit, janvier 1988 . [17] Séminaire de Géométrie Algébrique du Bois Marie , 1960 / 1961 , dirigé par A. GROTHENDIECK, Revêtements étales et groupe fondamental (Springer Lecture Notes in Mathematics, n^\circ 224, 1971 ). Zbl 0234.14002 · Zbl 0234.14002 [18] B. TEISSIER , Variétés polaires I : Invariants polaires des singularités d’hypersurfaces (Invent. Math., vol. 40, 1977 , p. 267-292). MR 57 #10004 | Zbl 0446.32002 · Zbl 0446.32002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.