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

Loeser, F.

Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 435-471 (1989).

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”.

Reviewer: M.Muro (Yanagido, Gifu)

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

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