eBook ISBN:  9781470429447 
Product Code:  MEMO/242/1145.E 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $50.40 
eBook ISBN:  9781470429447 
Product Code:  MEMO/242/1145.E 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $50.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 242; 2016; 131 ppMSC: Primary 14; 11; Secondary 52; 32; 58
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a nondegenerate surface singularity. The authors 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 nondegenerate 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 \(f^{1}(0)\subset\mathbf{C}^3\) 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 socalled \(B_1\)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 threedimensional 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 nondegenerate surface singularity.

Table of Contents

Chapters

1. Introduction

2. On the Integral Points in a ThreeDimensional 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 NonCompact $B_1$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 NonCompact $B_1$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 NonCompact $B_1$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 NonTrivial Character of $\mathbf {Z}_p^{\times }$

11. The Main Theorem in the Motivic Setting


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In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a nondegenerate surface singularity. The authors 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 nondegenerate 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 \(f^{1}(0)\subset\mathbf{C}^3\) 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 socalled \(B_1\)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 threedimensional 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 nondegenerate surface singularity.

Chapters

1. Introduction

2. On the Integral Points in a ThreeDimensional 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 NonCompact $B_1$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 NonCompact $B_1$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 NonCompact $B_1$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 NonTrivial Character of $\mathbf {Z}_p^{\times }$

11. The Main Theorem in the Motivic Setting