On rational singularities and counting points of schemes over finite rings

Glazer, Itay

doi:10.2140/ant.2019.13.485

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On rational singularities and counting points of schemes over finite rings

Itay Glazer

Vol. 13 (2019), No. 2, 485–500

DOI: 10.2140/ant.2019.13.485

Abstract

We study the connection between the singularities of a finite type $ℤ$ -scheme $X$ and the asymptotic point count of $X$ over various finite rings. In particular, if the generic fiber $X_{ℚ} = X \times_{Spec ℤ} Spec ℚ$ is a local complete intersection, we show that the boundedness of $| X (ℤ ∕ p^{n} ℤ) | ∕ p^{n dim X_{ℚ}}$ in $p$ and $n$ is in fact equivalent to the condition that $X_{ℚ}$ is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.

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Keywords

rational singularities, complete intersection, analysis on p-adic varieties, asymptotic point count

Mathematical Subject Classification 2010

Primary: 14B05

Secondary: 14G05

Milestones

Received: 3 March 2018

Revised: 27 August 2018

Accepted: 24 December 2018

Published: 2 March 2019

Authors

Itay Glazer
Faculty of Mathematics and Computer Science Weizmann Institute of Science Rehovot 7610001 Israel

Vol. 13, No. 2, 2019