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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 and Willem Veys

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: http://dx.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

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors
  • Chapter 3. Case I: Exactly One Facet Contributes to $s_0$ and this Facet Is a $B_1$-Simplex
  • Chapter 4. Case II: Exactly One Facet Contributes to $s_0$ and this Facet Is a Non-Compact $B_1$-Facet
  • Chapter 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
  • Chapter 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
  • Chapter 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
  • Chapter 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’
  • Chapter 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
  • Chapter 10. The Main Theorem for a Non-Trivial c Character of $\mathbf Z_p^\times $
  • Chapter 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 $f∈\mathbf Z[x,y,z]$ 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 $f^{-}1(0)⊂\mathbf C^{3}$ 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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