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 , and to exponential sums . For a -adic field , we denote by the ring of integers of and set the cardinal of the residue field. Let be a polynomial on and let be a character of . We define Igusa’s local zeta function associated to by
where is a Schwartz-Bruhat function and is the Haar measure on normalized that has measure 1. It is proved that is convergent if the real part is sufficiently large and is a rational function in .
In this survey, after stating some basic properties of this zeta function, the author gives the relation between the monodromy and the -function of , 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 -adic subanalytic sets and so on.