##
**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 \(\bmod p^ m\), and to exponential sums \(\bmod p^ m\). 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 \(f(x)\) be a polynomial on \(K^ n\) and let \(\chi\) be a character of \(R^ \times\). We define Igusa’s local zeta function associated to \(f(x)\) by \[ Z_ \Phi(s,\chi)=Z_ \Phi(s,\chi,K,f):=\int_{K^ n}\Phi(x)(acf(x))\;| f(x)|^ s\;| dx| \] where \(\Phi(x)\) 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 \(Z_ \Phi(s,\chi)\) is convergent if the real part \(\text{Re}(s)\) 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.

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 \(\bmod p^ m\), and to exponential sums \(\bmod p^ m\). 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 \(f(x)\) be a polynomial on \(K^ n\) and let \(\chi\) be a character of \(R^ \times\). We define Igusa’s local zeta function associated to \(f(x)\) by \[ Z_ \Phi(s,\chi)=Z_ \Phi(s,\chi,K,f):=\int_{K^ n}\Phi(x)(acf(x))\;| f(x)|^ s\;| dx| \] where \(\Phi(x)\) 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 \(Z_ \Phi(s,\chi)\) is convergent if the real part \(\text{Re}(s)\) 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.

Reviewer: M.Muro (Yanagido)

### MSC:

11S40 | Zeta functions and \(L\)-functions |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |

14M17 | Homogeneous spaces and generalizations |

14G20 | Local ground fields in algebraic geometry |