Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities: (Memoirs of the American Mathematical Society)

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.

Table of Contents:
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 s0s0 and this Facet Is a B1B1-Simplex Chapter 4. Case II: Exactly One Facet Contributes to s0s0 and this Facet Is a Non-Compact B1B1-Facet Chapter 5. Case III: Exactly Two Facets of ?f?f Contribute to s0s0, and These Two Facets Are Both B1B1-Simplices with Respect to a Same Variable and Have an Edge in Common Chapter 6. Case IV: Exactly Two Facets of ?f?f Contribute to s0s0, and These Two Facets Are Both Non-Compact B1B1-Facets with Respect to a Same Variable and Have an Edge in Common Chapter 7. Case V: Exactly Two Facets of ?f?f Contribute to s0s0; One of Them Is a Non-Compact B1B1-Facet, the Other One a B1B1-Simplex; These Facets Are B1B1 with Respect to a Same Variable and Have an Edge in Common Chapter 8. Case VI: At Least Three Facets of ?f?f Contribute to s0s0; All of Them Are B1B1-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 B1B1-Facets Contribute to s0s0; 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 Z×pZp× Chapter 11. The Main Theorem in the Motivic Setting

About the Author :
Bart Bories and Willem Veys, Katholieke Universiteit Leuven, Belgium.