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

Raf Cluckers, Itay Glazer and Yotam I. Hendel

Vol. 17 (2023), No. 12, 2229–2260

We provide uniform estimates on the number of pk-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 ), 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.

(FRS) morphisms, arc spaces, cell decomposition, counting points over finite rings, jet schemes, log-canonical threshold, motivic integration, p-adic integration, rational singularities, small ball estimates
Mathematical Subject Classification
Primary: 03C98, 11U09, 14B05, 14E18
Secondary: 11G25, 14G05
Received: 22 June 2022
Revised: 7 February 2023
Accepted: 20 March 2023
Published: 8 October 2023
Raf Cluckers
Laboratoire Painlevé
Université de Lille
KU Leuven
Department of Mathematics
Itay Glazer
Department of Mathematics
Northwestern University
United States
Yotam I. Hendel
Laboratoire Painlevé
Université de Lille

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