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 $𝜀$-jet flat morphisms, improving previous results by the last two authors.

我们对数量进行统一估计$\mathbb{Z}∕p^k\mathbb{Z}$-位于平滑簇之间的平坦态射纤维上的点，其纤维具有有理奇点，称为（FRS）态射。对于每根单独的纤维，估计值是通过 Avni 和 Aizenbud 的工作得知的，但我们使它们在所有纤维上保持一致。单个纤维的证明技术基于 Hironaka 的奇点解析和 Denef 公式，但在均匀情况下会失效。相反，我们使用动机整合理论的最新结果。此外，我们的估计相当于 (FRS) 属性，就像 Avni 和 Aizenbud 的绝对情况一样。此外，我们定义了新的态射类，称为$乙$- 平滑态射 ($乙\epsilon \mathbb{N}$），它细化了（FRS）属性，并使用我们开发的方法为其纤维提供如上所述的统一数论估计。对于纤维也给出了类似的估计$𝜀$-jet 平面态射，改进了最后两位作者之前的结果。