Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities

Bart Bories, Willem Veys (Autoren)

Buch | Softcover

131 Seiten

2016
American Mathematical Society (Verlag)
978-1-4704-1841-0 (ISBN)

Titel ist leider vergriffen;
keine Neuauflage

Artikel merken

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.

Bart Bories and Willem Veys, Katholieke Universiteit Leuven, Belgium.

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

Erscheinungsdatum	11.07.2016
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	218 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-1841-X / 147041841X
ISBN-13	978-1-4704-1841-0 / 9781470418410
Zustand	Neuware