Nagoya Mathematical Journal

Local zeta functions and Newton polyhedra

W. A. Zuniga-Galindo

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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 Néron $p$-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 $Z(s, f, \chi_{triv})$ in terms of $\Gamma(f)$ when the distance between $\Gamma(f)$ and the origin is at most one.

Article information

Nagoya Math. J., Volume 172 (2003), 31-58.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27]
Secondary: 11L05: Gauss and Kloosterman sums; generalizations


Zuniga-Galindo, W. A. Local zeta functions and Newton polyhedra. Nagoya Math. J. 172 (2003), 31--58.

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