Algebra & Number Theory

On the normalized arithmetic Hilbert function

Mounir Hajli

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Abstract

Let X ¯N be a subvariety of dimension n, and let norm(X; ) be the normalized arithmetic Hilbert function of X introduced by Philippon and Sombra. We show that this function admits the asymptotic expansion

norm(X;D) = ĥ(X) (n + 1)!Dn+1 + o(Dn+1),D 1,

where ĥ(X) is the normalized height of X. This gives a positive answer to a question raised by Philippon and Sombra.

Article information

Source
Algebra Number Theory, Volume 9, Number 10 (2015), 2293-2302.

Dates
Received: 19 November 2014
Revised: 10 September 2015
Accepted: 15 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842446

Digital Object Identifier
doi:10.2140/ant.2015.9.2293

Mathematical Reviews number (MathSciNet)
MR3437762

Zentralblatt MATH identifier
1352.14015

Subjects
Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 11G50: Heights [See also 14G40, 37P30] 11G35: Varieties over global fields [See also 14G25]

Keywords
arithmetic Hilbert function height

Citation

Hajli, Mounir. On the normalized arithmetic Hilbert function. Algebra Number Theory 9 (2015), no. 10, 2293--2302. doi:10.2140/ant.2015.9.2293. https://projecteuclid.org/euclid.ant/1510842446


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References

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