Uniform estimate of Lang-Weil type for counting points of schemes over finite rings
Abstract
In this paper, we study the number of points of ℤ-schemes X of finite type over the finite rings ℤ/pmℤ for positive integers m and prime numbers p. Motivated by Igusa's conjecture for exponential sums and the Lang-Weil estimate, we establish an estimate of X(ℤ/pmℤ) in terms of m, X(ℤ/pm−1ℤ) and poles of the p-adic Igusa local zeta function of X uniformly in m≥ 2 and p. Our estimate might be viewed as an inductive version for counting points of schemes over finite rings of the Lang-Weil estimate. Our estimate agrees with the expected bound of Igusa's conjecture for exponential sums if Xℚ is not a locally complete intersection having only rational singularities. This can be regarded as an analogous version of the estimate of exponential sums modulo powers of primes of polynomials with non-rational singularities given by Cluckers, Mustaţǎ and the author in 2019. In the case that Xℚ is a locally complete intersection having only rational singularities, our estimate is very close to the expected bound of Igusa's conjecture for exponential sums. To prove our estimate, beside using the usual tools such as resolutions of singularities and Denef's formula for Igusa local zeta functions, we also use the Hermit interpolation formula to study the numerators of Igusa local zeta functions. By the same method, we are able to approach Igusa's conjecture for exponential sums in the case of rational singularities. Lastly, to have further applications of our result in the future, we study heavily on p-adic analysis to give an upper bound of the real parts of non-trivial poles of Igusa local zeta functions in terms of suitable generalized log canonical thresholds.
Related
- Counting points on Hessenberg varieties over finite fields
- Non-reducedness of the Hilbert schemes of few points
- Hochschild cohomology of Hilbert schemes of points on surfaces
- Nef cone decompositions for nested Hilbert schemes of points on surfaces
- The canonical class and Chern character operators on Hilbert schemes of points
BibTeX
@misc{nguyen2026uniform,
title = {{Uniform estimate of Lang-Weil type for counting points of schemes over finite rings}},
author = {Kien Huu Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01748},
eprint = {2610.01748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01748},
}
RIS
TY - UNPB
TI - Uniform estimate of Lang-Weil type for counting points of schemes over finite rings
AU - Kien Huu Nguyen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01748
UR - https://arxiv.org/abs/2610.01748
AB - In this paper, we study the number of points of ℤ-schemes X of finite type over the finite rings ℤ/p^mℤ for positive integers m and prime numbers p. Motivated by Igusa's conjecture for exponential sums and the Lang-Weil estimate, we establish an estimate of X(ℤ/p^mℤ) in terms of m, X(ℤ/p^m−1ℤ) and poles of the p-adic Igusa local zeta function of X uniformly in m≥ 2 and p. Our estimate might be viewed as an inductive version for counting points of schemes over finite rings of the Lang-Weil estimate. Our estimate agrees with the expected bound of Igusa's conjecture for exponential sums if X_ℚ is not a locally complete intersection having only rational singularities. This can be regarded as an analogous version of the estimate of exponential sums modulo powers of primes of polynomials with non-rational singularities given by Cluckers, Mustaţǎ and the author in 2019. In the case that X_ℚ is a locally complete intersection having only rational singularities, our estimate is very close to the expected bound of Igusa's conjecture for exponential sums. To prove our estimate, beside using the usual tools such as resolutions of singularities and Denef's formula for Igusa local zeta functions, we also use the Hermit interpolation formula to study the numerators of Igusa local zeta functions. By the same method, we are able to approach Igusa's conjecture for exponential sums in the case of rational singularities. Lastly, to have further applications of our result in the future, we study heavily on p-adic analysis to give an upper bound of the real parts of non-trivial poles of Igusa local zeta functions in terms of suitable generalized log canonical thresholds.
ER -