## Local zeta functions and Euler characteristics.(English)Zbl 0738.11060

This paper deals with poles of Igusa’s local zeta functions. Let $$K$$ be a number field, $${\mathcal O}_ K$$ its integer ring and $$\mathfrak p$$ be a maximal ideal of $${\mathcal O}_ K$$. For a polynomial $$f(x)$$ on $$K^ m$$, we define the local zeta function $$Z(K_{\mathfrak p},\phi,s)=\int_{\mathbb{R}^ m_{\mathfrak p}}\phi(ac(f(x))) | f(x)|^ s| dx|$$, where $$K_{\mathfrak p}$$ and $$R_{\mathfrak p}$$ are $$\mathfrak p$$-completions of $$K$$ and $${\mathcal O}_ K$$, respectively and $$\phi(ac(\;))$$ is a character, $$|\;|$$ means $$\mathfrak p$$-adic absolute value, $$| dx|$$ is the Haar measure on $$\mathbb{R}^ m_{\mathfrak p}$$. The author’s main result is the following. We give it without precise explanation on the notations. Let $$\phi$$ be a character of $$\mathbb{R}^*_{\mathfrak p}$$ of order $$d$$ which is trivial on $$1+{\mathfrak p}\mathbb{R}_{\mathfrak p}$$ and assume $${\mathfrak p}\in\mathbb{Q}$$, $$\rho<0$$, and put $$T_{\rho,d}=\{i\in T\mid -\nu_ i/N_ i=\rho, d| N_ i, \bar E_ i\neq\emptyset\}$$. Suppose for each $$i \in T_{\rho,d}$$ that $$\bar E_ i$$ is proper and that there is no $$E_ j$$ intersecting $$E_ i$$ with $$d\mid N_ j$$, $$j\neq i$$. Then we have the following two properties. (1) If $$\chi(E^ 0_ i)=0$$ for all $$i\in T_{\rho,d}$$, then $$Z(K_{\mathfrak p},\phi,s)$$ has no pole with real part $$\rho$$. (2) If $$\chi(E^ 0_ i)\neq 0$$ for some $$i\in T_{\rho,d}$$, then there are infinitely many unramified extensions $$L_{\mathfrak p}$$ of $$K_{\mathfrak p}$$ such that $$\rho$$ is a pole of $$Z(K_{\mathfrak p},\phi\circ N_{L_{\mathfrak p}/K_{\mathfrak p}},s)$$, where $$N_{L_{\mathfrak p}/K_{\mathfrak p}}$$ denotes the norm. Here, $$E_ i$$’s are irreducible components of a resolution of $$f(x)=0$$ with normal crossing, and $$N_ i$$, $$\nu_ i$$ are multiplicities concerning them.
Reviewer: M.Muro (Yanagido)

### MSC:

 11S40 Zeta functions and $$L$$-functions

### Keywords:

poles; Igusa’s local zeta functions
Full Text:

### References:

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