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Non-archimedean canonical measures on abelian varieties

Part of:Arithmetic problems. Diophantine geometryArithmetic algebraic geometry

Published online by Cambridge University Press:  21 April 2010

Walter Gubler
Walter Gubler*
Affiliation:
Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany (email: walter.gubler@uni-tuebingen.de)
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Abstract

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For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures c1(L∣X)∧d on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford’s construction. The results have applications to the equidistribution of small points.

Keywords

Berkovich analytic spacesformal geometryChambert-Loir’s measurestropical varietiesequidistribution

MSC classification

Secondary:14G40: Arithmetic varieties and schemes; Arakelov theory; heights11G10: Abelian varieties of dimension $> 1$14G22: Rigid analytic geometry

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010, pp. 683 - 730
Copyright
Copyright © Foundation Compositio Mathematica 2010

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