Igusaâe(tm)s

Cover -- Title page -- Chapter 0. Introduction -- 0.1. Igusa's zeta function and the Monodromy Conjecture -- 0.2. Statement of the main theorem -- 0.3. Preliminaries on Newton polyhedra -- 0.4. Theorems of Denef and Hoornaert -- 0.5. Expected order and contributing faces -- 0.6. ₁-facets and the structure of the proof of the main theorem -- 0.7. Overview of the paper -- Acknowledgments -- Chapter 1. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors -- 1.0. Introduction -- 1.1. A group structure on -- 1.2. Divisibility among the multiplicities , ₁, ₂, ₃ -- 1.3. On the ₁ points of ₁ -- 1.4. On the ₁-coordinates of the points of -- 1.5. More divisibility relations -- 1.6. Explicit description of the points of -- 1.7. Determination of the numbers ₁, ₂, ₃, , ', ₀ from the coordinates of ₁, ₂, ₃ -- Chapter 2. Case I: Exactly One Facet Contributes to ₀ and this Facet Is a ₁-Simplex -- 2.1. Figure and notations -- 2.2. Some relations between the variables -- 2.3. Igusa's local zeta function -- 2.4. The candidate pole ₀ and its residue -- 2.5. Terms contributing to ₁ -- 2.6. The numbers _{ } -- 2.7. The factors _{ }( ₀) -- 2.8. Multiplicities of the relevant simplicial cones -- 2.9. The sums Σ(⋅)( ₀) -- 2.10. A new formula for ₁ -- 2.11. Formulas for Σ_{ } and Σ_{ } -- 2.12. A formula for _{ }=\mult _{ } -- 2.13. Description of the points of _{ } -- 2.14. A formula for Σ_{ } -- 2.15. Proof of ₁'=0 -- Chapter 3. Case II: Exactly One Facet Contributes to ₀ and this Facet Is a Non-Compact ₁-Facet -- 3.1. Figure and notations -- 3.2. The candidate pole ₀ and the contributions to its residue -- 3.3. The factors _{ }( ₀), the sums Σ(⋅)( ₀) and a new formula for ₁ -- 3.4. Proof of ₁'=0.

Chapter 4. Case III: Exactly Two Facets of Γ_{ } Contribute to ₀, and These Two Facets Are Both ₁-Simplices with Respect to a Same Variable and Have an Edge in Common -- 4.1. Figure and notations -- 4.2. Some relations between the variables -- 4.3. Igusa's local zeta function -- 4.4. The candidate pole ₀ and its residues -- 4.5. Terms contributing to ₂ and ₁ -- 4.6. The numbers _{ } -- 4.7. The factors _{ }( ₀) and _{ }'( ₀) -- 4.8. Multiplicities of the relevant simplicial cones -- 4.9. The sums Σ(⋅)( ₀) and Σ(⋅)'( ₀) -- 4.10. Simplified formulas for ₂ and ₁ -- 4.11. Vector identities -- 4.12. Points of _{ }, _{ }, _{ }, ₂, ₁ and additional relations -- 4.13. Investigation of the Σ_{∙} and the Σ_{∙}', except for Σ₁', Σ₃ -- 4.14. Proof of ₂=0 and a new formula for ₁ -- 4.15. Study of Σ₁' -- 4.16. An easier formula for the residue ₁ -- 4.17. Investigation of Σ₃ -- 4.18. Proof that the residue ₁ equals zero -- Chapter 5. Case IV: Exactly Two Facets of Γ_{ } Contribute to ₀, and These Two Facets Are Both Non-Compact ₁-Facets with Respect to a Same Variable and Have an Edge in Common -- 5.1. Figure and notations -- 5.2. The candidate pole ₀ and the contributions to its residues -- 5.3. Towards simplified formulas for ₂ and ₁ -- 5.4. Some vector identities and their consequences -- 5.5. Points of ₓ, _{ }, and ₃ -- 5.6. Formulas for Σₓ,Σₓ', and Σ₃ -- 5.7. Proof of ₂'= ₁'=0 -- Chapter 6. Case V: Exactly Two Facets of Γ_{ } Contribute to ₀ -- One of Them Is a Non-Compact ₁-Facet, the Other One a ₁-Simplex -- These Facets Are ₁ with Respect to a Same Variable and Have an Edge in Common -- 6.1. Figure and notations -- 6.2. Contributions to the candidate pole ₀ -- 6.3. Towards simplified formulas for ₂ and ₁ -- 6.4. Investigation of the sums Σ_{∙} and Σ_{∙}' -- 6.5. Proof of ₂'= ₁'=0.

Chapter 7. Case VI: At Least Three Facets of Γ_{ } Contribute to ₀ -- All of Them Are ₁-Facets (Compact or Not) with Respect to a Same Variable and They Are 'Connected to Each Other by Edges' -- Chapter 8. General Case: Several Groups of ₁-Facets Contribute to ₀ -- Every Group Is Separately Covered By One of the Previous Cases, and the Groups Have Pairwise at Most One Point in Common -- Chapter 9. The Main Theorem for a Non-Trivial Character of _{ }^{×} -- Chapter 10. The Main Theorem in the Motivic Setting -- 10.1. The local motivic zeta function and the motivic Monodromy Conjecture -- 10.2. A formula for the local motivic zeta function of a non-degenerate polynomial -- 10.3. A proof of the main theorem in the motivic setting -- References -- Back Cover.

In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate 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 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)\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 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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