Measures and equidistribution in Berkovich spaces. (Mesures et équidistribution sur les espaces de Berkovich.) (French) Zbl 1112.14022

The author proves the non-archimedean analogue of a theorem of L. Szpiro, E. Ullmo and S. Zhang [Invent. Math. 127, No. 2, 337–347 (1997; Zbl 0991.11035)] on equidistribution of small points, namely, if a general sequence of geometrical points on a projective variety over a number field satisfies \(h_{\bar{L}} (x_n) \rightarrow h_{\bar{L}}(X)\) (so the points are “small”) where \(\bar{L}\) is an ample line bundle with semi-positive adelic metric and \(h_{\bar{L}}\) the associated height, then if \(v\) is a place of \(F\) for which the \(L\)-metric is ample, the sequence of Dirac measures concentrated at the points \(x_n\) and their Galois conjugates converge weakly (to a known probability measure) on the Berkovich space of \(X\) at \(v\). For curves, there is a similar stronger result where one can drop the condition of ampleness on the \(L\)-metric. Finally, an application is given to equidistribution on the reduction graph of Chinburg-Rumely. As an example: for an elliptic curve with semistable reduction, points of small height are equidistributed for the rotation invariant measure on the reduction (which is a circle).


14G22 Rigid analytic geometry
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G25 Global ground fields in algebraic geometry


Zbl 0991.11035
Full Text: DOI arXiv


[1] Bouche A. Abbes, Fourier (Grenoble) 45 pp 2– (1995)
[2] Autissier P., Math. 531 pp 201– (2001)
[3] DOI: 10.1155/S1073792804130183 · Zbl 1080.14027
[4] Yu, Duke Math. J. 89 pp 3– (1997)
[5] S. Bosch, Math. Ann. 295 pp 2– (1993)
[6] DOI: 10.2307/2152736 · Zbl 0973.14013
[7] S. David, Contemp. Math. 210 pp 333– (1998)
[8] DOI: 10.2307/2007043 · Zbl 0559.14005
[9] Soulé H. Gillet, Paris Sér. I Math. 307 pp 887– (1988)
[10] DOI: 10.1007/BF02699132 · Zbl 0741.14012
[11] DOI: 10.1007/BF01231343 · Zbl 0777.14008
[12] Gubler W., Symp. Math. 37 pp 190– (1997)
[13] Gubler W., Math. 498 pp 61– (1998)
[14] Gubler W., Ann. Scu. Norm. Sup. Pisa 2 pp 4– (2003)
[15] Maillot V., Mém. Soc. Math. France 80 pp 129– (2000)
[16] DOI: 10.1215/S0012-7094-99-09701-6 · Zbl 1161.11325
[17] DOI: 10.1007/s002220050123 · Zbl 0991.11035
[18] DOI: 10.2307/2946601 · Zbl 0788.14017
[19] DOI: 10.1007/BF01232429 · Zbl 0795.14015
[20] DOI: 10.2307/2152886 · Zbl 0861.14018
[21] Zhang S.-W., J. Alg. Geom. 4 pp 281– (1995)
[22] DOI: 10.2307/120986 · Zbl 0991.11034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.