$E$-光滑与(FRS)态射的数论刻画：$\u2124/p^{k}\u2124$-点数量的估计
A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points
AI总结：
该研究利用动机积分理论，给出了(FRS)态射纤维上$\u2124/p^{k}\u2124$-点数量的一致估计，证明其与(FRS)性质等价，还定义了细化的$E$-光滑态射并给出对应估计，同时改进了$\u03b5$-射流平坦态射的相关结果。
摘要：
我们对光滑代数簇之间平坦态射的纤维上的$\u2124/p^{k}\u2124$-点数量给出了一致估计，这类态射的纤维具有有理奇点，被称为(FRS)态射。对于单条纤维，相关估计已由Avni和Aizenbud的工作得到，但我们的结果在所有纤维上具有一致性。单纤维情形的证明技术基于Hironaka奇点消解和Denef公式，但该方法在一致情形下失效。为此，我们采用了来自动机积分理论的最新成果。此外，我们的估计与(FRS)性质等价，这与Avni和Aizenbud在绝对情形下的结论一致。另外，我们定义了一类新的态射，称为$E$-光滑态射（$E\u2208\u2115$），它是对(FRS)性质的细化，并利用我们发展的方法为其纤维提供了上述一致数论估计。我们还对$\u03b5$-射流平坦态射的纤维给出了类似估计，改进了最后两位作者此前的结果。
英文摘要：
We provide uniform estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka's resolution of singularities and Denef's formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called $E$-smooth morphisms ($E\in\mathbb{N}$), which refine the (FRS) property, and use the methods we developed to provide uniform number-theoretic estimates as above for their fibers. Similar estimates are given for fibers of $\varepsilon$-jet flat morphisms, improving previous results by the last two authors.
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