*(French)*Zbl 0749.11054

[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 ${R}^{\times}$. We define Igusa’s local zeta function associated to $f\left(x\right)$ by

where ${\Phi}\left(x\right)$ is a Schwartz-Bruhat function and $\left|dx\right|$ 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 $\text{Re}\left(s\right)$ 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 |