Title: Igusa's p-adic local zeta function and the Monodromy Conjecture for non-degenerate surface singularities
Authors: Bories, Bart ×
Veys, Wim #
Issue Date: 2016
Publisher: The Society
Series Title: Memoirs of the American Mathematical Society vol:242 issue:1145 pages:1-+
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 in 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 the complex zero locus of f 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.
ISSN: 0065-9266
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Algebra Section
× corresponding author
# (joint) last author

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