Arithmetic of Higher-Dimensional Algebraic Varieties
One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on curves, gleaned in part through the theorems of Mordell, Weil, Siegel, and Faltings. It has become clear that
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Arithmetic of Higher-Dimensional Algebraic Varieties
Bjorn Poonen Yuri Tschinkel Editors
Springer Science+Business Media, LLC
Bjom Poonen Department of Mathematics University of California, Berkeley Berkeley, CA 94720 U.S.A.
Yuri Tschinkel Mathematisches Institut Bunsenstr. 3-5 D-37073 Gottingen Germany
Library of Congress Cataloging-in-Publication Data
A CW catalogue record for this book is available from the Library of Congress, Washington D.C., USA.
AMS Subject Classifications: 11045, 11D72, llF72, 11G35, llG50, 11Y50, 14D72, 14E30, 14G05, 14G25
ISBN 978-1-4612-6471-2 ISBN 978-0-8176-8170-8 (eBook) DOI 10.1007/978-0-8176-8170-8
Printed on acid-free paper.
@2004 Springer Science+Business Media New York Originally published by Birkhauser Boston in 2004 Softcover reprint ofthe hardcover lst edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not 10 be taken as an expression of opinion as 10 whether or not they are subject 10 property rights. 987654321
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SPIN 10936412
CONTENTS
Abstracts
Xl
Introduction
xv Part I.
E xpository Articles
SIR P ET ER SWIN NERTON- DvER Diophantine equati ons: progress an d problems 1. Int rodu ction 2. The Hasse P rinciple and th e Brauer- Manin obst ruction 3. Zet a functions and L-series 4. Curves 5. Generalities about surfaces 6. Rational surfaces 7. K3 surfaces and Kum mer surfaces 8. Density of rationa l points References RO GER H EAT H-B ROW N -
Rationa l points and analytic numb er theory
1. Introduction
2. Cont ributions from an alyt ic number t heory 3. Potenti al applications to analytic num ber th eory References Weak approximation on algebraic vari eties 1. Introductio n 2. Classical resu lts 3. Cohomological methods References
D AV ID H A R A RI -
3 3 6 12 15 19 20 27 28 32
.. 37 37 38 40 41
43 43 44 50 57
CON TENTS
vi
Count ing points on varieties using universal torsors . . . . . .. .. . .. . . . . ... . . .. ... .. .. ... . . . .. . . . . . .. . . . . . . .. . . . . . . . ... . . .. . .... . . 1. Introduction 2. Heights on projective varieties 3. Manin 's principl e 4. Results 5. The counterexample of Batyrev and Tschinkel 6. Methods of counting 7. A basic example 8. Universal torsors 9. Toric varieties 10. Th e plan e blown up in 4 points 11. Generalization References
E MMANUEL PEYRE -
Part II.
61 61 63 66 70 71 71 72 73 76 78 79 80
Research Articles
& OLEG N. POPOV - The Cox ring of a Del Pe zzo surface 85 1. Introduction 85 2. Del Pezzo
