Algebra & Number Theory

On the normalized arithmetic Hilbert function

Mounir Hajli

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

arithmetic Hilbert function height


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

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  • A. Abbes and T. Bouche, “Théorème de Hilbert–Samuel `arithmétique”', Ann. Inst. Fourier $($Grenoble$)$ 45:2 (1995), 375–401.
  • F. Amoroso and S. David, “Minoration de la hauteur normalisée dans un tore”, J. Inst. Math. Jussieu 2:3 (2003), 335–381.
  • S. David and P. Philippon, “Minorations des hauteurs normalisées des sous-variétés des tores”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 28:3 (1999), 489–543.
  • H. Gillet and C. Soulé, “An arithmetic Riemann–Roch theorem”, Invent. Math. 110:3 (1992), 473–543.
  • P. Philippon and M. Sombra, “Hauteur normalisée des variétés toriques projectives”, J. Inst. Math. Jussieu 7:2 (2008), 327–373.
  • H. Randriambololona, Hauteurs pour les sous-schémas et exemples d'utilisation de méthodes arakeloviennes en théorie de l'approximation diophantienne, Ph.D. thesis, Université Paris-Sud, 2001, hook \posturlhook.
  • H. Randriambololona, “Métriques de sous-quotient et théorème de Hilbert–Samuel arithmétique pour les faisceaux cohérents”, J. Reine Angew. Math. 590 (2006), 67–88.
  • S. Zhang, “Positive line bundles on arithmetic surfaces”, Ann. of Math. $(2)$ 136:3 (1992), 569–587.
  • S. Zhang, “Small points and adelic metrics”, J. Algebraic Geom. 4:2 (1995), 281–300.