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On the poles of Igusa’s local zeta function for curves. (English) Zbl 0659.14016
Let F be a number field, A its ring of algebraic integers, and $${\mathfrak P}$$ any maximal ideal of A. Denote by $$\hat A_{{\mathfrak P}}$$ and $$\hat F_{{\mathfrak P}}$$ the $${\mathfrak P}$$-adic completion of A and F, respectively. - Let $$f\in F[x,y]$$; Igusa’s local zeta function of f is defined by $Z(s)=\int_{\hat A^ 2_{{\mathfrak P}}}| f(x,y)|^ s| dx\wedge dy|$ for $$s\in {\mathbb{C}}$$, $$Re(s)>0$$. A well-known list of candidate poles of (the meromorphic continuation to $${\mathbb{C}}$$ of) Z(s) is given as follows.
Let $$\Pi$$ : $$X\to \hat F^ 2_{{\mathfrak P}}$$ be an embedded resolution of $$f=0$$ and $$E_ j$$, $$j\in T$$, the irreducible components of $$\Pi^{- 1}(f^{-1}\{0\})$$ with numerical data $$(N_ j,\nu_ j)$$. Then all real poles of Z(s) can be expressed as $$s=-\nu_ j/N_ j$$, $$j\in T$$. - Let $$s_ 0\in \{-\nu_ j/N_ h| j\in T\}$$ not be induced by a component of the strict transform of $$f=0$$. Then, for almost all $${\mathfrak P}$$, we show: $$s_ 0$$ is a pole of Z(s) if and only if at least one $$E_ j$$ with $$s_ 0=-\nu_ j/N_ j$$ occurs such that $$E_ j$$ intersects the remaining components of $$\Pi^{-1}(f^{-1}\{0\})$$ in at least 3 points.
Reviewer: W.Veys

MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11S40 Zeta functions and $$L$$-functions
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