Igusa's conjecture for exponential sums

Prove that there exist positive constants \(c\) and \(M\), depending only on a non-constant polynomial \(f\) and a finite-type subscheme \(Z\), such that \(|E_{f,L,Z,\psi}|\leq c\,m_\psi^{n-1}q_L^{-m_\psi\,\operatorname{moi}_{K,Z}(f)}\) for every sufficiently large-residue-characteristic non-Archimedean local field \(L\) and every additive character \(\psi\) of conductor \(m_\psi\geq2\).

Background

The conjecture gives a uniform bound for exponential sums in terms of the motivic oscillation index. It is central to the paper's motivation because such bounds imply estimates for point counts over finite rings and have applications to local-global principles.

The paper proves the conjecture in several cases, including situations involving non-rational singularities and obtains weaker or modified bounds in other cases. The displayed conjecture itself is stated as the general unresolved target.

References

Then there are positive constants $c,M$ depending only on $f,Z$ such that $$\left|E_{f,L,Z,\psi}\right|\leq cm_{\psi}{n-1}q_L{-m_\psimoi_{K,Z}(f)}$$ for all local fields $L\in\tilde{}{K,M}$ and all additive characters $\psi$ of $L$ of conductor $m{\psi}\geq 2$.

— Uniform estimate of Lang-Weil type for counting points of schemes over finite rings  (2610.01748 - Nguyen, 1 Oct 2026) in Section 2, Subsection 2.3.1, Conjecture 2.3.1 (labelled Igusa's conjecture for exponential sums)

For each positive integer $r$, there is an integer $M$ depending on $, Z$ and a positive constant $c_r$ depending on $,r$ such that for all local fields $L\in\tilde{}{K,M}$ and all $m\geq 2$, we have $$\left|E{L,Z,}{(r)}(m)\right|\leq c_rm{n+r-1}q_L{-mmoi_{K,Z}{(r)}()}.$$

— Uniform estimate of Lang-Weil type for counting points of schemes over finite rings  (2610.01748 - Nguyen, 1 Oct 2026) in Section 2, Subsection 2.3.2, Conjecture 2.3.2 (labelled Averaged Igusa conjecture for exponential sums)