A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points

TL;DR

This paper establishes uniform estimates for the number of points on fibers of certain morphisms between smooth varieties, linking these estimates to properties like (FRS) and introducing new classes called $E$-smooth morphisms.

Contribution

It provides the first uniform estimates for fibers of (FRS) morphisms using motivic integration and introduces $E$-smooth morphisms, refining the existing (FRS) property.

Findings

Uniform point-count estimates for (FRS) fibers.

Equivalence of estimates and (FRS) property.

Introduction of $E$-smooth morphisms with refined estimates.

Abstract

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.