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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

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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