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 $\bmod p\sp m$, and to exponential sums $\bmod p\sp m$. 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 $K\sp n$ and let $\chi$ be a character of $R\sp \times$. We define Igusa’s local zeta function associated to $f(x)$ by $$Z\sb \Phi(s,\chi)=Z\sb \Phi(s,\chi,K,f):=\int\sb{K\sp n}\Phi(x)(acf(x))\ \vert f(x)\vert\sp s\ \vert dx\vert$$ where $\Phi(x)$ is a Schwartz-Bruhat function and $\vert dx\vert$ is the Haar measure on $K\sp n$ normalized that $R\sp n$ has measure 1. It is proved that $Z\sb \Phi(s,\chi)$ is convergent if the real part $\text{Re}(s)$ is sufficiently large and is a rational function in $q\sp{-s}$.
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 $f(x)\sp s$, 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.