Igusa’s 𝑝-adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities
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Publication:5365187
DOI10.1090/MEMO/1145zbMATH Open1397.14020arXiv1306.6012OpenAlexW2963739945MaRDI QIDQ5365187
Author name not available (Why is that?)
Publication date: 6 October 2017
Published in: (Search for Journal in Brave)
Abstract: In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f in Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of the complex zero locus of f close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B_1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerated surface singularity.
Full work available at URL: https://arxiv.org/abs/1306.6012
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