Work

On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities

Year: 2026

Type: preprint

Abstract: In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over $p$-adic fields with good reduction. In this paper, by slightly generalizing the $p$-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of $\mathbb{Q}$-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.12222

Authors: Nguyen, Kien Huu, Yasuda, Takehiko

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Vlaamse regering, Région Normandie, Japan Society for the Promotion of Science

Awards: Interaction of singularities and number theory, Number theory and algebraic geometry and computer mathematics - New developments on rationality problems for algebraic varieties -

OpenAlex ID: W7221182365

Cites: 3

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Related to: 10