Cycle Classes in Anabelian Geometry
Cycle classes appear in the context of anabelian geometry in different incarnations, see for example Parshin (The Grothendieck Festschrift, vol. 3, 1990), Mochizuki (Invent. Math. 138(2):319–423, 1999; Mathematical Sciences Research Institute Publications
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Jakob Stix
Rational Points and Arithmetic of Fundamental Groups Evidence for the Section Conjecture
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Jakob Stix Mathematics Center Heidelberg (MATCH) University of Heidelberg Heidelberg, Germany
ISBN 978-3-642-30673-0 ISBN 978-3-642-30674-7 (eBook) DOI 10.1007/978-3-642-30674-7 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012945519 Mathematics Subject Classification (2010): 14H30, 14G05, 14H25, 11G20, 14G32, 14F35 c Springer-Verlag Berlin Heidelberg 2013 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Antonia, Jaden and Lucie and to Sabine
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Preface
The section conjecture, as stated by Grothendieck [Gr83] in a letter to Faltings in 1983, speculates about a representation of rational points in the realm of anabelian geometry. Every k-rational point of a geometrically connected variety X=k gives rise to a conjugacy class of sections s W 1 .Spec.k// ! 1 .X / of the natural projection 1 .X / ! 1 .Spec.k// of e´ tale fundamental groups. The section conjecture suggests that the converse also holds for smooth, projective curves of genus at least 2 over
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