发表机构
KU Leuven; Institute of Mathematics, Vietnam Academy of Science and Technology（荷语鲁汶大学; 越南科学技术研究院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限环上概形点数问题，建立了与Igusa猜想和Lang-Weil估计一致的一致估计，并利用Hermite插值及p-adic分析处理奇点情形。
AI 中文摘要
本文研究了有限环$\mathbb{Z}/p^m\mathbb{Z}$上有限型$\mathbb{Z}$-概形$X$的点数，其中$m$为正整数，$p$为素数。受Igusa指数和猜想及Lang-Weil估计的启发，我们建立了$X(\mathbb{Z}/p^m\mathbb{Z})$关于$m$、$X(\mathbb{Z}/p^{m-1}\mathbb{Z})$以及$X$的$p$-进Igusa局部zeta函数的极点的估计，该估计对$m\geq 2$和$p$一致成立。我们的估计可视为Lang-Weil估计在有限环上概形点数问题中的归纳版本。若$X_{\mathbb{Q}}$不是仅具有有理奇点的局部完全交集，则我们的估计与Igusa指数和猜想的预期界一致。这可视为Cluckers、Mustaţǎ和作者于2019年给出的关于具有非有理奇点的多项式模素数幂的指数和估计的类似版本。在$X_{\mathbb{Q}}$是仅具有有理奇点的局部完全交集的情形下，我们的估计非常接近Igusa指数和猜想的预期界。为证明我们的估计，除使用奇点消解和Igusa局部zeta函数的Denef公式等常用工具外，我们还使用Hermite插值公式来研究Igusa局部zeta函数的分子。通过相同的方法，我们能够在有理奇点情形下逼近Igusa指数和猜想。最后，为使我们的结果在未来有进一步应用，我们深入研究了$p$-进分析，以给出Igusa局部zeta函数的非平凡极点的实部在适当的广义对数典范阈值方面的上界。
英文摘要
In this paper, we study the number of points of $\mathbb{Z}$-schemes $X$ of finite type over the finite rings $\mathbb{Z}/p^m\mathbb{Z}$ 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 $X(\mathbb{Z}/p^m\mathbb{Z})$ in terms of $m$, $X(\mathbb{Z}/p^{m-1}\mathbb{Z})$ and poles of the $p$-adic Igusa local zeta function of $X$ uniformly in $m\geq 2$ 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 $X_{\mathbb{Q}}$ 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 $X_{\mathbb{Q}}$ 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.
Comments66 pages. Comments welcome
