Local Zeta Function for Curves, Non Degeneracy Conditions and Newton Polygons

2001/07/26 by M. J. Saia, Saia, M. J., W. A. Zuniga-Galindo +1
Mathematics · #11D79 #14G20 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11D79 #msc:14G20 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/0107189

32 pages, 4 figures

arxiv created 2003/07/03 · arxiv updated 2009/11/30

Abstract

This paper is dedicated to the description of the poles of the Igusa local zeta functions Z(s,f,v) when f(x,y) satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial f(x,y), a collection of convex sets Γ^A called the arithmetic Newton polygon of f(x,y), and introduce the notion of arithmetic non degeneracy with respect to Γ^A(f). The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials with respect to some arithmetic Newton polygon.If L is a number field, our main result asserts that for almost all non-archimedean valuations v of L, the poles of Z(s,f,v), with f(x,y)∈ L[x,y], can be described explicitly in terms of the equations of the straight segments that conform the boundaries of the convex sets that belong to Γ^A(f). Moreover, our proof gives an effective procedure to compute Z(s,f,v).

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• Igusa's Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials
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