Wonderful compactifications of Bruhat-Tits buildings

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

Non-archimedean canonical measures on abelian varieties

For a closedd-dimensional subvarietyXof an abelian varietyAand a canonically metrized line bundleLonA, Chambert-Loir has introduced measuresc1(L∣X)∧don the Berkovich analytic space associated toAwith 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 ofAand Mumford’s construction. The results have applications to the equidistribution of small points.