Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization
The formalism presented in this monograph has been developed with a view toward applications to Diophantine geometry and transcendence theory. In Diophantine geometry, one of ten encounters situations that combine formal geometry over the integers and com
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Jean-Benoît Bost
Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves
Progress in Mathematics Volume 334
Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA
More information about this series at http://www.springer.com/series/4848
Jean-Benoît Bost
Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves
Jean-Benoît Bost Département de Mathématique Université Paris-Sud Orsay, France
ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-030-44328-3 ISBN 978-3-030-44329-0 (eBook) https://doi.org/10.1007/978-3-030-44329-0 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dilectissimae uxori necnon consilio optimae
Contents
Preface
xi
Introduction 1
1
Hermitian Vector Bundles over Arithmetic Curves 1.1 Definitions and Basic Operations . . . . . . . . . . . . . . . . . . . 1.2 Direct Images. The Canonical Hermitian Line Bundle ω OK /Z over Spec OK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Arakelov Degree and Slopes . . . . . . . . . . . . . . . . . . . . . . 1.4 Morphisms and Extensions of Hermitian Vector Bundles . . . . . .
11 11 13 14 16
2 θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves 2.1 The Poisson Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The θ-Invariants h0θ and h1θ and the Poiss
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