Multivariate normal distribution for integral points on varieties

Daniel El-Baz; Daniel Loughran; Efthymios Sofos

doi:10.1090/tran/8545

Multivariate normal distribution for integral points on varieties

Daniel El-Baz, Daniel Loughran, Efthymios Sofos

Department of Mathematical Sciences

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Abstract

Abstract: Given a variety with coefficients in , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

Original language	English
Pages (from-to)	3089-3128
Number of pages	40
Journal	Transactions of the American Mathematical Society
Volume	375
Issue number	5
Early online date	24 Feb 2022
DOIs	https://doi.org/10.1090/tran/8545
Publication status	Published - 31 Dec 2022

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.1090/tran/8545

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First published in Trans. Amer. Math. Soc 375 (2022), published by the American Mathematical Society. © 2022 American Mathematical Society.
Accepted author manuscript, 528 KBLicence: CC BY-NC-ND

Cite this

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title = "Multivariate normal distribution for integral points on varieties",

abstract = "Abstract: Given a variety with coefficients in , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erd{\H o}s–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution. ",

author = "Daniel El-Baz and Daniel Loughran and Efthymios Sofos",

note = "Funding Information: Received by the editors September 6, 2020, and, in revised form, May 27, 2021. 2020 Mathematics Subject Classification. Primary 14G05; Secondary 60F05, 11N36. The first author was supported by the Austrian Science Fund (FWF), projects F-5512 and Y-901. The second author was supported by EPSRC grant EP/R021422/2. ",

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AU - Loughran, Daniel

AU - Sofos, Efthymios

N1 - Funding Information: Received by the editors September 6, 2020, and, in revised form, May 27, 2021. 2020 Mathematics Subject Classification. Primary 14G05; Secondary 60F05, 11N36. The first author was supported by the Austrian Science Fund (FWF), projects F-5512 and Y-901. The second author was supported by EPSRC grant EP/R021422/2.

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Y1 - 2022/12/31

N2 - Abstract: Given a variety with coefficients in , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

AB - Abstract: Given a variety with coefficients in , we study the distribution of the number of primes dividing the coordinates as we vary an integral point. Under suitable assumptions, we show that this has a multivariate normal distribution. We generalise this to more general Weil divisors, where we obtain a geometric interpretation of the covariance matrix. For our results we develop a version of the Erdős–Kac theorem that applies to fairly general integer sequences and does not require a positive exponent of level of distribution.

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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Multivariate normal distribution for integral points on varieties

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Quantitative arithmetic geometry

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Multivariate normal distribution for integral points on varieties

Abstract

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Quantitative arithmetic geometry

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