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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 $\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$, and to exponential sums $\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{m}$. 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 $f\left(x\right)$ be a polynomial on ${K}^{n}$ and let $\chi$ be a character of ${R}^{×}$. We define Igusa’s local zeta function associated to $f\left(x\right)$ by

${Z}_{{\Phi }}\left(s,\chi \right)={Z}_{{\Phi }}\left(s,\chi ,K,f\right):={\int }_{{K}^{n}}{{\Phi }\left(x\right)\left(acf\left(x\right)\right)\phantom{\rule{4pt}{0ex}}|f\left(x\right)|}^{s}\phantom{\rule{4pt}{0ex}}|dx|$

where ${\Phi }\left(x\right)$ 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 ${Z}_{{\Phi }}\left(s,\chi \right)$ is convergent if the real part $\text{Re}\left(s\right)$ 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 $f{\left(x\right)}^{s}$, 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.

##### MSC:
 11S40 Zeta functions and $L$-functions of local number fields 11-02 Research monographs (number theory) 14G10 Zeta-functions and related questions 32S40 Monodromy; relations with differential equations and $D$-modules 14M17 Homogeneous spaces and generalizations 14G20 Local ground fields