Exponential sums and motivic oscillation index of arbitrary ideals and their applications
Kien Huu Nguyen
TL;DR
The paper develops a bridge between arithmetic and geometry by introducing abstract exponential sums modulo $p^m$ for arbitrary ideals and defining the motivic oscillation index $ ext{moi}$, viewing it as the arithmetic counterpart to geometric invariants like the Bernstein–Sato polynomial and the monodromy. Central to the work is the $r$-th exponential sum $E_{\mathcal{I}}^{(r)}(p,m)$ and its uniform bounds governed by $ ext{moi}_{K,Z}^{(r)}(\mathcal{I})$, together with the Averaged Igusa conjecture which posits uniform decay in $m$ for large primes. The text develops the theory through log-resolutions, jet schemes, and zeta functions, and then ties these invariants to concrete applications: counting points over finite rings, properties of FRS/FTS/FGI morphisms, and Waring-type problems, all underpinned by explicit bounds and transfer principles. The results illuminate how arithmetic thresholds such as $ ext{lct}$ and poles of Igusa zeta functions reflect deep geometric features like rational and semi-log canonical singularities, and they propose precise conjectures to leverage these connections in broad contexts. Overall, the work provides a robust framework to study arithmetic of ideals via motivic invariants, with substantial consequences for uniform counting, morphism regularity, and additive combinatorics in number-theoretic settings.
Abstract
In 2006, Budur, Mustaţǎ and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. Because of the strong monodromy conjecture, it should have a corresponding picture on the arithmetic side of ideals in polynomial rings. In this paper, we try to address this problem. Motivated by the Hardy-Littlewood circle method, we introduce the notions of abstract exponential sums modulo $p^m$ and motivic oscillation index of an arbitrary ideal in polynomial rings over number fields. In the arithmetic picture, the abstract exponential sums modulo $p^m$ and the motivic oscillation index of an ideal should play the role of the Bernstein-Sato polynomial and its maximal non-trivial root of the corresponding scheme. We will provide some properties of the motivic oscillation index of ideals in this paper. On the other hand, based on Igusa's conjecture for exponential sums, we propose the averaged Igusa conjecture for exponential sums of ideals. In particular, this conjecture and the motivic oscillation index of ideals will have many interesting applications. We will introduce these applications and prove some variant version of this conjecture.