On the motivic oscillation index and bound of exponential sums modulo $p^m$ via analytic isomorphisms

TL;DR

This paper explores the motivic oscillation index of polynomials, investigates its stability and relation to Igusa's conjecture, and proves the conjecture for specific classes of polynomials using advanced algebraic and model-theoretic techniques.

Contribution

It advances understanding of the motivic oscillation index, links it to Igusa's conjecture, and proves the conjecture for polynomials of certain types, including three-variable and ADE singularities.

Findings

Proved Igusa's conjecture for polynomials in three variables.

Established positivity results for poles of twisted Igusa's local zeta functions.

Linked motivic oscillation index stability to conjectures on $\, ext{l}$-adic cohomology weights.

Abstract

Let $f$ be a polynomial in $n$ variables over some number field and $Z$ a subscheme of affine $n$-space. The notion of motivic oscillation index of $f$ at $Z$ was initiated by Cluckers (2008) and Cluckers-Musta\c{t}\v{a}-Nguyen (2019). In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta functions of $f$ has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of $\ell$-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if $f$ is a polynomial of Thom-Sebastiani type with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of $ADE$ type. In an appendix, we answer affirmatively a recent question of Cluckers-Musta\c{t}\v{a}-Nguyen (2019) on poles of twisted Igusa's local zeta functions of maximal order.