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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
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