Abstract
We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.
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References
Autissier P. (2006) Équidistribution de sous-variétés de petite hauteur. J. Théor. Nombres Bordx. 18(1): 1–12
Baker M., Ih S.-I. (2004) Equidistribution of small subvarieties of an abelian variety. N. Y. J. Math. 10: 279–289
Berkovich V.G. (1990) Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. AMS, Providence
Bilu Y. (1997) Limit distribution of small points on algebraic tori. Duke Math. J. 89(3): 465–476
Bombieri E., Gubler W. (2006) Heights in Diophantine Geometry. Cambridge University Press, Cambridge
Bosch S., Güntzer U., Remmert R. (1984) Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundl. Math. Wiss., vol. 261. Springer, Berlin
Bosch S., Lütkebohmert W. (1993) Formal and rigid geometry. I: Rigid spaces. Math. Ann. 295(2): 291–317
Chambert-Loir A. (2006) Mesure et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595: 215–235
Faber, X.W.C.: Equidistribution of dynamically small subvarieties over the function field of a curve, arXiv:math.NT:0801.4811v2 (preprint)
Gubler W. (1998) Local heights of subvarieties over non-archimedean fields. J. Reine Angew. Math. 498: 61–113
Gubler W. (2003) Local and canonical heights of subvarieties. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (Ser. V) 2(4): 711–760
Gubler W. (2007) Tropical varieties for non-archimedean analytic spaces. Invent. Math. 169(2): 321–376
Gubler W. (2007) The Bogomolov conjecture for totally degenerate abelian varieties. Invent. Math. 169(2): 377–400
Gubler, W.: Non-archimedean canonical measures on abelian varieties, available at arXiv:math.NT:0801.4503v1 (preprint)
Lang S. (1958) Introduction to Algebraic Geometry. Interscience Publishers, New York
Lazarsfeld R. (2004) Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Bd. 48. Springer, Berlin
Szpiro L., Ullmo E., Zhang S. (1997) Equirépartition des petits points. Invent. Math. 127: 337–347
Ullmo E. (1998) Positivité et discrétion des points algébriques des courbes. Ann. Math. (2) 147(1): 167–179
Ullrich P. (1995) The direct image theorem in formal and rigid geometry. Math. Ann. 301(1): 69–104
Yuan, X.: Positive line bundles over arithmetic varieties, available at arXiv:math. NT:0612424v1 (preprint)
Zhang S. (1998) Equidistribution of small points on abelian varieties. Ann. Math. (2) 147(1): 159–165