تقرير
A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points
| العنوان: | A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^{k}\mathbb{Z}$-points |
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| المؤلفون: | Cluckers, Raf, Glazer, Itay, Hendel, Yotam I. |
| المصدر: | Alg. Number Th. 17 (2023) 2229-2260 |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Logic, Mathematics - Number Theory, 03C98, 11U09, 14E18, 14B05 (Primary) 11G25, 14G05 (Secondary) |
| الوصف: | 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. Comment: 27 pages, comments welcome; v2: the new notion of E-smooth morphisms was added, and uniform estimates on the number of points lying on the fibers of $E$-smooth and $\varepsilon$-jet flat morphisms are given (Theorems 4.11 and 4.12) |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.2140/ant.2023.17.2229 |
| URL الوصول: | http://arxiv.org/abs/2103.00282 |
| رقم الأكسشن: | edsarx.2103.00282 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.2140/ant.2023.17.2229 |
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