[Paper Review] Igusa's p-adic local zeta function and the Monodromy Conjecture for non-degenerated surface singularities

Bart Bories, Willem Veys|arXiv (Cornell University)|Jun 25, 2013

Algebraic Geometry and Number Theory25 references7 citations

TL;DR

This paper establishes the Monodromy Conjecture for Igusa’s p-adic and motivic zeta functions of non-degenerate surface singularities by proving that all poles of the p-adic zeta function correspond to eigenvalues of the local monodromy. The authors resolve the key challenge of showing that candidate poles arising from $B_1$-facets in the Newton polyhedron are not actual poles, using a detailed combinatorial analysis of integral points in 3D fundamental parallelepipeds and residue vanishing arguments, building on Lemahieu and Van Proeyen’s earlier topological result.

ABSTRACT

In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated 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-degenerated 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-degenerated surface singularity.

Motivation & Objective

• To extend the Monodromy Conjecture from the topological to the p-adic and motivic setting for non-degenerate surface singularities.
• To resolve the central challenge that candidate poles from $B_1$-facets in the Newton polyhedron are not actual poles.
• To establish that every pole of Igusa’s p-adic zeta function corresponds to an eigenvalue of the local monodromy operator.
• To provide a complete proof of the Monodromy Conjecture for these zeta functions using combinatorial and algebraic techniques.

Proposed method

• Analyzes Igusa’s p-adic local zeta function via resolution of singularities and motivic integration techniques.
• Studies integral points in 3-dimensional fundamental parallelepipeds spanned by primitive vectors to control residue contributions.
• Applies divisibility arguments in the ring $\mathbf{Z}[S,S^{-1}][T]$ to verify that certain factors in the denominator do not contribute to poles.
• Uses Puiseux series over $\overline{\mathbf{Q}}$ to test root conditions for polynomial divisibility in the motivic setting.
• Performs case-by-case residue vanishing analysis across six configurations of $B_1$-facets (compact and non-compact).
• Leverages prior work by Lemahieu and Van Proeyen on the topological zeta function to transfer results to the p-adic and motivic settings.

Experimental results

Research questions

• RQ1 Do all poles of Igusa’s p-adic zeta function for non-degenerate surface singularities correspond to eigenvalues of the local monodromy operator?
• RQ2 Are candidate poles arising from $B_1$-facets in the Newton polyhedron actually poles, or do they vanish due to residue cancellation?
• RQ3 How can the residue vanishing condition be proven combinatorially for $B_1$-facets in three dimensions?
• RQ4 Can the Monodromy Conjecture be extended from the topological to the p-adic and motivic zeta functions for surface singularities?
• RQ5 What is the precise structure of the motivic zeta function in terms of $B_1$-facet contributions and their associated factors?

Key findings

• All poles of Igusa’s p-adic zeta function for non-degenerate surface singularities correspond to eigenvalues of the local monodromy operator at the origin.
• Candidate poles arising from $B_1$-facets are not actual poles, as their residues vanish due to deep combinatorial cancellation in the integral point structure of 3D fundamental parallelepipeds.
• The proof establishes that the motivic zeta function’s denominator factors associated with $B_1$-facets divide the numerator, ensuring no spurious poles.
• The residue vanishing is verified via Puiseux series root analysis in $\overline{\mathbf{Q}}\{\{S\}\/}$, confirming divisibility in $\mathbf{Z}[S,S^{-1}][T]$.
• The result extends the Monodromy Conjecture to the p-adic and motivic zeta functions, completing the conjecture for non-degenerate surface singularities.
• The analysis covers all possible configurations of $B_1$-facets (up to five types), including compact and non-compact cases, and their interactions via shared edges.