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Preprint Number 1926

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1926. Raf Cluckers, Itay Glazer and Yotam I. Hendel
A number theoretic characterization of (FRS) morphisms: estimates on the number of ℤ/p^kℤ-points

Submission date: 27 February 2021


We provide estimates on the number of ℤ/p^kℤ-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.

Mathematics Subject Classification: 03C98, 11U09, 14E18, 14B05 (Primary) 11G25, 14G05 (Secondary)

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Full text arXiv 2103.00282: pdf, ps.

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