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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 modp m , and to exponential sums modp 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 χ be a character of R × . We define Igusa’s local zeta function associated to f(x) by

Z Φ (s,χ)=Z Φ (s,χ,K,f):= K n Φ(x)(acf(x))|f(x)| s |dx|

where Φ(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 Φ (s,χ) is convergent if the real part 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.

11S40Zeta functions and L-functions of local number fields
11-02Research monographs (number theory)
14G10Zeta-functions and related questions
32S40Monodromy; relations with differential equations and D-modules
14M17Homogeneous spaces and generalizations
14G20Local ground fields