## 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).

### MSC:

 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
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### References:

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