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Poles of a local zeta function and Newton polygons. (English) Zbl 0606.14022
Let K be a non-archimedean field of characteristic zero, R the ring of integers and f a polynomial in n variables with coefficients in R. Using the Haar measure \(d\mu\) on the additive group of K, Igusa defined a zeta function associated to f which extended the classical Mellin transform introduced by Gel’fand-Shilov over the reals and complexes. - For s a complex number with positive real part and \(\phi\) a locally constant complex valued function on \(K^ n\), define the zeta function by \(Z_ f(s,\phi)=\int | f|^ s\phi d\mu \) where the integration is over \(K^ n\). In the archimedean setting, with \(\phi\) a Schwartz-Bruhat function, it was established, using Hironaka’s resolution of singularities, that this function has a meromorphic continuation with poles contained in finitely many arithmetic progressions of negative rationals. Igusa observed that these proofs could be adopted in the non- archimedean case and showed that \(Z_ f(s,\phi)\) was extendible to a rational function in \(q^{-s}\) where q is the cardinality of the residue field.
The current article continues the investigations begun by D. Meuser in Invent. Math. 73, 445-465 (1983; Zbl 0512.14015) to detect the poles of the continuation. The basic strategy is to work with a Hironaka resolution of f which determines a collection of negative rational numbers that contains the poles of the extension. However, in the known examples most of these numbers are not poles. The point here is to determine the actual poles. This is accomplished for a certain class of reducible plane curves with exactly one singularity at (0,0) which is ”toroidal”. These singularities have the property that a resolution can be constructed from information derived from the Newton polygon associated to a defining equation for the curve. Moreover, the poles depend on the singularity and can also be described in terms of the Newton polygon. A determination of the largest pole is given in terms of the polygon, and is identical to the description in the archimedean cases. The negative of this value also admits an interpretation in terms of mixed Hodge structures.
Reviewer: Lawrence G.Roberts

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G25 Global ground fields in algebraic geometry
11S40 Zeta functions and \(L\)-functions
32P05 Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32-XX describing the type of problem)
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