## Report on Igusa’s local zeta function.(French)Zbl 0749.11054

Séminaire Bourbaki, Vol. 1990/91, Exp. No. 730-744, Astérisque 201-203, 359-386 (1991).
[For the entire collection see Zbl 0742.00056).]
This paper is a survey work on the recent development of studies on Igusa’s local zeta functions and related topics. It is closely related to the number of solutions of congruences $$\bmod p^ m$$, and to exponential sums $$\bmod p^ m$$. For a $$p$$-adic field $$K$$, we denote by $$R$$ the ring of integers of $$K$$ and set $$q$$ the cardinal of the residue field. Let $$f(x)$$ be a polynomial on $$K^ n$$ and let $$\chi$$ be a character of $$R^ \times$$. We define Igusa’s local zeta function associated to $$f(x)$$ by $Z_ \Phi(s,\chi)=Z_ \Phi(s,\chi,K,f):=\int_{K^ n}\Phi(x)(acf(x))\;| f(x)|^ s\;| dx|$ where $$\Phi(x)$$ is a Schwartz-Bruhat function and $$| dx|$$ is the Haar measure on $$K^ n$$ normalized that $$R^ n$$ has measure 1. It is proved that $$Z_ \Phi(s,\chi)$$ is convergent if the real part $$\text{Re}(s)$$ is sufficiently large and is a rational function in $$q^{-s}$$.
In this survey, after stating some basic properties of this zeta function, the author gives the relation between the monodromy and the $$b$$-function of $$f(x)^ s$$, and some functional equations involved in them. The latter half of this article is devoted to the explanation of some special topics such as prehomogeneous vector spaces, the integration on $$p$$-adic subanalytic sets and so on.
Reviewer: M.Muro (Yanagido)

### MSC:

 11S40 Zeta functions and $$L$$-functions 11-02 Research exposition (monographs, survey articles) pertaining to number theory 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 14M17 Homogeneous spaces and generalizations 14G20 Local ground fields in algebraic geometry

Zbl 0742.00056
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