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Igusa’s $p$-adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities
About this Title
Bart Bories, Department of Mathematics, KU Leuven, Celestijnenlaan 200b – box 2400, 3001 Leuven, Belgium and Willem Veys, Department of Mathematics, KU Leuven, Celestijnenlaan 200b – box 2400, 3001 Leuven, Belgium
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 242, Number 1145
ISBNs: 978-1-4704-1841-0 (print); 978-1-4704-2944-7 (online)
DOI: https://doi.org/10.1090/memo/1145
Published electronically: February 29, 2016
Keywords: Monodromy Conjecture,
Igusa’s zeta function,
motivic zeta function,
surface singularity,
non-degenerate,
lattice polytope
MSC: Primary 14D05, 11S80, 11S40, 14E18, 14J17; Secondary 52B20, 32S40, 58K10
Table of Contents
Chapters
- 1. Introduction
- 2. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors
- 3. Case I: Exactly One Facet Contributes to $s_0$ and this Facet Is a $B_1$-Simplex
- 4. Case II: Exactly One Facet Contributes to $s_0$ and this Facet Is a Non-Compact $B_1$-Facet
- 5. Case III: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both $B_1$-Simplices with Respect to a Same Variable and Have an Edge in Common
- 6. Case IV: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both Non-Compact $B_1$-Facets with Respect to a Same Variable and Have an Edge in Common
- 7. Case V: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$; One of Them Is a Non-Compact $B_1$-Facet, the Other One a $B_1$-Simplex; These Facets Are $B_1$ with Respect to a Same Variable and Have an Edge in Common
- 8. Case VI: At Least Three Facets of $\Gamma _f$ Contribute to $s_0$; All of Them Are $B_1$-Facets (Compact or Not) with Respect to a Same Variable and They Are ’Connected to Each Other by Edges’
- 9. General Case: Several Groups of $B_1$-Facets Contribute to $s_0$; Every Group Is Separately Covered By One of the Previous Cases, and the Groups Have Pairwise at Most One Point in Common
- 10. The Main Theorem for a Non-Trivial Character of $\mathbf {Z}_p^{\times }$
- 11. The Main Theorem in the Motivic Setting
Abstract
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate 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 $f\in \mathbf {Z}[x,y,z]$ satisfying $f(0,0,0)=0$ and non-degenerate 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 $f^{-1}(0)\subseteq \mathbf {C}^3$ 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-degenerate surface singularity.
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