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Motivic Igusa zeta functions. (English) Zbl 0943.14010

Let \(p\) be a prime number, let \(K\) be a finite extension of \({\mathbb Q}_p\) with valuation ring \(R\), maximal ideal \(P\) of \(R\), and residue field \(\overline{K}=R/P\simeq{\mathbb F}_q\). For \(z\in K\) write \(|z|=q^{-\text{ord} z}\), where \(\text{ord} z\) is the valuation of \(z\). For a non-constant element \(f\in K[x_1,\ldots,x_m]\) one defines the \(p\)-adic Igusa local zeta function \(Z(s)\) associated to \(f\) and with trivial character as the \(p\)-adic integral \({Z(s):=\int_{R^m}|f(x)|^s|dx|}\), where \(|dx|\) is the Haar measure on \(K^m\) (normalized such that \(R^m\) has volume \(1\)), and where \(s\in{\mathbb C}\), \(\operatorname{Re}(s)>0\). More generally, for a multiplicative character \(\alpha:R^\times\to{\mathbb C}^\times\), for \(\pi\) a fixed uniformizing parameter of \(R\) and with \(\text{ac}(z):=z\pi^{-\text{ord} z}\), \(z\in K\), one defines the \(p\)-adic Igusa local zeta function \({Z(s,\alpha):=\int_{R^m}\alpha(\text{ac}(f(x)))|f(x)|^s|dx|}\). For a field \(k\) of characteristic zero consider the Grothendieck ring \(K_0(\text{Sch}_k)\) generated by symbols \([S]\) for algebraic varieties \(S\) over \(k\). In particular, let \({\mathbf L}=[{\mathbb A}_k^1]\). As an analogue to \(Z(s)\) one may construct a series \(Z_{\text{geom}}(s)\in K_0(\text{Sch}_k)[{\mathbf L}^{-1}][[{\mathbf L}^{-s}]]\) which for a fixed value of \(s\in{\mathbb N}\) can be interpreted as a Kontsevich integral. To extend this definition to the case of \(Z(s,\alpha)\) for a non-trivial character \(\alpha\), one considers a smooth connected separated scheme \(X\) of finite type over \(k\), and a multiplicative character \(\alpha\) of any finite subgroup of \(k^\times\). For a reduced subscheme \(W\) of \(X\) and a morphism \(f:X\to{\mathbb A}^1_k\), one defines the motivic Igusa function \({\int_W(f^s,\alpha)\in K_0({\mathcal M})[[{\mathbf L}^{-s}]]}\) where \(K_0({\mathcal M})\) is the Grothendieck ring of Chow motives and \(\mathbf L\) is the Lefschetz motive. It is shown that a motivic Igusa function is rational, and in case \(W=X={\mathbb A}_k^m\) and \(f\) a homogeneous polynomial, \({\int_X(f^s,\alpha)}\), \(\alpha\) trivial, satisfies a functional equation of the form \({\left(\int_Xf^s\right)^{\vee}={\mathbf L}^{-rs}\int_Xf^s\in K_0({\mathcal M}_k)[{\mathbf L}^s,{\mathbf L}^{-s}]_{\text{loc}}}\), where \({\int_Xf^s=\int_X(f^s,\text{triv})}\). For general \(\alpha\) one should replace the Grothendieck group of Chow motives by the Grothendieck group of Voevodsky’s triangulated category of geometrical motives. One may relate motivic Igusa functions to nearby cycles by considering \({{{\mathbf L}^m}\over{1-{\mathbf L}}}\lim_{s\to -\infty}\int_{\{x\}}(f^s,\alpha)\), where \(x\) is a closed point of the fiber \(f^{-1}(0)\).

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14F42 Motivic cohomology; motivic homotopy theory
11S40 Zeta functions and \(L\)-functions
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