Titlebook: Arakelov Geometry over Adelic Curves; Huayi Chen,Atsushi Moriwaki Book 2020 Springer Nature Singapore Pte Ltd. 2020 Arakelov geometry.Adel

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Arakelov Geometry over Adelic Curves978-981-15-1728-0Series ISSN 0075-8434 Series E-ISSN 1617-9692

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Metrized vector bundles: local theory,fundament for the global study of adelic vector bundles. Note that we need to consider both Archimedean and non-Archimedean cases. Hence we carefully choose the approach of presentation to unify the statements whenever possible, and to clarify the differences.

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Adelic curves,47] in the number field setting. This theory allows to consider all places of a global field in a unified way. It also leads to a uniform approach in the geometry of numbers in global fields, either via the adelic version of Minkowski’s theorems and Siegel’s lemma developed by McFeat [105], Bombieri

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Slopes of tensor product,recisely, give a family . of adelic vector bundles over a proper adelic curve ., we give a lower bound of . in terms of the sum of the minimal slopes of . minus a term which is the product of three half of the measure of the infinite places and the sum of ., see Corollary 5.6.2 for details. This res

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,Nakai-Moishezon’s criterion,is dense in each .., where . ∈ .. We let .. be the set of all . ∈ . such that |⋅|. is the trivial absolute value. Note that, if .. is not empty, then the above hypothesis implies that, either the .-algebra . is discrete, or the field . is countable.

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