On rational singularities and counting points of schemes over finite rings

Itay Glazer

doi:10.2140/ant.2019.13.485

On rational singularities and counting points of schemes over finite rings

Itay Glazer^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

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 _xSpecℤ Specℚ is a local complete intersection, we show that the boundedness of |X(ℤ/pⁿℤ)|/p^ndimX _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.

Original language	English (US)
Pages (from-to)	485-500
Number of pages	16
Journal	Algebra and Number Theory
Volume	13
Issue number	2
DOIs	https://doi.org/10.2140/ant.2019.13.485
State	Published - Mar 2 2019

Funding

I would like to thank my advisor Avraham Aizenbud for presenting me with this problem, teaching and helping me in this work. I hold many thanks to Nir Avni for helpful discussions and for hosting me at Northwestern University in July 2016, during which a large part of this work was done. I also thank Yotam Hendel for fruitful talks. This work was partially supported by the ISF grant [687/13], the BSF grant [2012247] and the Minerva Foundation grant.

Keywords

Analysis on p-adic varieties
Asymptotic point count
Complete intersection
Rational singularities

ASJC Scopus subject areas

Algebra and Number Theory

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10.2140/ant.2019.13.485

Cite this

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title = "On rational singularities and counting points of schemes over finite rings",

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 Xℚ = X xSpecℤ Specℚ is a local complete intersection, we show that the boundedness of |X(ℤ/pnℤ)|/pndimX ℚ in p and n is in fact equivalent to the condition that Xℚ is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.",

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N2 - 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 Xℚ = X xSpecℤ Specℚ is a local complete intersection, we show that the boundedness of |X(ℤ/pnℤ)|/pndimX ℚ in p and n is in fact equivalent to the condition that Xℚ is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.

AB - 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 Xℚ = X xSpecℤ Specℚ is a local complete intersection, we show that the boundedness of |X(ℤ/pnℤ)|/pndimX ℚ in p and n is in fact equivalent to the condition that Xℚ is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.

KW - Analysis on p-adic varieties

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KW - Rational singularities

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On rational singularities and counting points of schemes over finite rings

Abstract

Funding

Keywords

ASJC Scopus subject areas

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