Rigid Geometry of Curves and Their Jacobians
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The
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Werner Lütkebohmert
Rigid Geometry of Curves and Their Jacobians
Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge
A Series of Modern Surveys in Mathematics
Editorial Board L. Ambrosio, Pisa V. Baladi, Paris cedex 05 G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette G. Huisken, Tübingen J. Jost, Leipzig J. Kollár, Princeton S.S. Kudla, Toronto G. Laumon, Orsay Cedex U. Tillmann, Oxford J. Tits, Paris D.B. Zagier, Bonn
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Volume 61
Werner Lütkebohmert
Rigid Geometry of Curves and Their Jacobians
Werner Lütkebohmert Institute of Pure Mathematics Ulm University Ulm, Germany
ISSN 0071-1136 ISSN 2197-5655 (electronic) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-319-27369-3 ISBN 978-3-319-27371-6 (eBook) DOI 10.1007/978-3-319-27371-6 Library of Congress Control Number: 2016931305 Mathematics Subject Classification (2010): 14G22, 14H40, 14K15, 30G06 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2016 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
En hommage à Michel Raynaud
Preface
Projective algebraic curves or abelian varieties are defined as the vanishing locus of finite families of homogeneous polynomials in a projective space fulfilling certain conditions. Except for elliptic curves or hyperelliptic curves, it is difficult to pin down equations which give rise to curves or abelian varieties. Over the complex numbers one has analytic tools to construct and to uniformize such objects. For example, every smooth curve of genus g ≥ 2 has a representation Γ \H, where H is the upper half-plane and Γ ⊂ Aut(H) is a group acting on H. Similarly, every compact complex Lie group is of type Cn /Λ, where Λ is a lattice in Cn ; the abelian varieties among the compact complex Lie groups can be char
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