Title :

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

Author(s) :

Cluckers, Raf [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Glazer, Itay [Auteur]
Hendel, Yotam I. [Auteur]

Journal title :

Algebra & Number Theory

Pages :

2029-2060

Publisher :

Mathematical Sciences Publishers

Publication date :

2023-10-08

ISSN :

1937-0652

English keyword(s) :

(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

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL