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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 |
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Vol. 17 (2023), No. 12, 2229–2260
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Abstract |
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We provide uniform estimates on the number of -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 -smooth morphisms (), 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 -jet flat morphisms, improving previous results by the last two authors. |
Keywords
(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
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Mathematical Subject Classification
Primary: 03C98, 11U09, 14B05, 14E18
Secondary: 11G25, 14G05
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Milestones
Received: 22 June 2022
Revised: 7 February 2023
Accepted: 20 March 2023
Published: 8 October 2023
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Authors |
| Raf Cluckers | |
| Laboratoire Painlevé Université de Lille Lille France |
KU Leuven Department of Mathematics Leuven Belgium |
| Itay Glazer | |
| Department of Mathematics Northwestern University Evanston IL United States |
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| Yotam I. Hendel | |
| Laboratoire Painlevé Université de Lille Lille France |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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