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

Chambert-Loir, Antoine

J. Reine Angew. Math. 595, 215-235 (2006).

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).

Reviewer: Gunther Cornelissen (Utrecht)

MSC:

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

Keywords:

Berkovich Space, Arakelov Geometry, non-archimedean measure, equidistribution

Citations:

Zbl 0991.11035

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