Local zeta functions and Newton polyhedra

Details Local zeta functions and Newton polyhedra Local zeta functions and Newton polyhedra

2002-04-19

arxiv.org/abs/math/0204241v2

Mathematics

Algebraic Geometry

26 pages, revised version, accepted for publication in Nagoya Math. J

Scientific paper

To a polynomial $f$ over a non-archimedean local field $K$ and a character $\chi$ of the group of units of the valuation ring of $K$ one associates Igusa's local zeta function $Z(s,f,\chi)$. In this paper, we study the local zeta function $Z(s,f,\chi)$ associated to a non-degenerate polynomial $f$, by using an approach based on the p-adic stationary phase formula and N\'eron p-desingularization. We give a small set of candidates for the poles of $Z(s,f,\chi)$ in terms of the Newton polyhedron $ \Gamma(f)$ of $f$. We also show that for almost all $\chi$, the local zeta function $Z(s,f,\chi)$ is a polynomial in $q^{-s}$ whose degree is bounded by a constant independent of $\chi$. Our second result is a description of the largest pole of $Z(s,f, \chi_{\text{triv}})$ in terms of $ \Gamma(f)$ when the distance between $\Gamma(f)$ and the origin is at most one.

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