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Introduction to Arakelov Theory BOOKS of INTEREST by SERGE LANG Introduction to Arakelov Theory BOOKS OF INTEREST BY SERGE LANG Geometry: A High School Course (with Gene Morrow) This high school text, inspired by a researcher's educational interests and Gene Morrow's experience as a high school teacher, presents geometry to the student in an exemplary, accessible, and attractive form. The book emphasizes both the intel­ lectually stimulating parts of geometry, including physical and classical applications, and routine arguments or computations. MATH! Encounters with High School Students This book is a faithful record of dialogues between Lang and high school students, covering some of the topics in the Geometry book and others of the same mathemat­ ical level (areas and volumes of classical figures, pythagorean triples, infinities). These encounters have been transcribed from tapes, and thus are true, authentic, and alive. The Beauty of Doing Mathematics: Three Public Dialogues Here we have three dialogues given in three successive years between Lang and au­ diences at the Palais de la Decouverte (science museum) in Paris. The audiences consisted of many types of persons, including some high school students, but pri­ marily people who just drop in at the Palais on a Saturday afternoon. Lang did mathematics with them: prime numbers, elliptic curves, and Thurston's conjecture on the classification of 3-manifolds. Again the encounters were taped, recording Lang's success at putting a lay audience in contact with basic research problems of mathematics. This was done by using a selection of topics which could be explained from scratch in a self-contained way. The enthusiasm of the audience is partly evi­ denced by the fact that, the third time, more than 100 persons stayed for three and one-half hours, dealing with the Thurston material. Fundamentals of Diophantine Geometry A systematic account of fundamentals, including the basic theory of heights, Roth and Siegel's theorems, the Neron-Tate quadratic form, the Mordell-Weil theorem, Weil and Neron functions, and the canonical form on a curve as it relates to the Ja­ cobian via the theta functions. Introduction to Complex Hyperbolic Spaces Since its introduction by Kobayashi, the theory of complex hyperbolic spaces has progressed considerably. This book gives an account of some of the most important results, such as Brody's theorem, hyperbolic imbeddings, curvature properties, and some Nevalinna theory. It also includes Cartan's proof for the Second Main Theo­ rem, which was elegant and short. * OTHER BOOKS BY LANG PUBLISHED BY SPRINGER-VERLAG Introduction to Arakelov Theory· Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Elliptic Curves: Dio­ phantine Analysis· Modular Units (with Daniel Kubert) • Introduction to Algebraic and Abelian Functions· Cyclotomic Fields • Elliptic Functions • Algebraic Num­ ber Theory· SL2(R) • Abelian Varieties • Differential Manifolds· Complex Analysis· Undergraduate Analysis • Undergraduate Algebra· Linear Algebra • Introduction to Linear Algebra· Calculus of Several Variables· First Course in Calculus • Basic Mathematics· THE FILE Serge Lang Introduction to Arakelov Theory Springer Science+Business Media, LLC Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A. Mathematics Subject Classifications (1980): 11G05, 11010 Library of Congress Cataloging-in-Publication Data Lang, Serge Introduction to Arakelov theory I Serge Lang. p. em. Bibliography: p. Includes index. ISBN 978-1-4612-6991-5 ISBN 978-1-4612-1031-3 (eBook) DOI 10.1007/978-1-4612-1031-3 1. Arithmetical algebraie geometry. 1. Title. II. Title: Arakelov theory. QA242.5.L36 1988 512'.72--de 19 88-15952 © 1988 by Springer Science+Business MediaNew York Originally published by Springer-Verlag New York Inc. in 1985 Softcover reprint ofthe hardcover Ist edition 1985 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe 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 general descriptive names, trade names, trademarks, etc. in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Typeset by Composition House Ltd., Salisbury, England. 9 8 7 6 5 4 3 2 1 ISBN 978-1-4612-6991-5 Foreword The purpose of this book is to give an introduction to Arakelov theory. This theory consists in transposing results of algebraic geometry, especially numerical results, to the number field case, by completing the set of points (or divisors) to include those at infinity, that is, the archi­ medean places. This is done by considering hermitian structures in addi­ tion to sheaf structures, so analysis becomes number theory at infinity. One can then get the analogous results in a number of cases. For a curve over the ring of integers of a number field, one gets for instance the adjunction formula; the Hodge Index Theorem, treated here as in Hriljac's paper, which means identifying the Arakelov intersection numbers with the Neron-Tate form and using the positive definiteness of this form rather than reproving it; and the Faltings Riemann-Roch theorem on arithmetic surfaces. The proof of Riemann·-Roch here de­ pends on the adjunction formula, but does not depend on semistability assumptions which are made systematically in Faltings, and subsequent expositions (sometimes only implicitly). Faltings' other results on the Noether formula are much deeper; they depend on the moduli spaces, and I did not feel I could include them in this book. I have, however, included Faltings' theorem on the positivity of the dualizing sheaf in the semistable case. The Riemann-Roch theorem has a formulation not only in terms of a numerical relationship but in terms of a metric isomorphism of certain sheaves, due to Faltings for metrics with the canonical Chern form. Deligne [De] has pointed to the possibility of dealing with arbitrary metrics. I have not entered into this aspect of the theorem. On the other hand, the residue theorem and the discussion of the metrics involved may be viewed as carrying out this more general study for vi FOREWORD adjunction. Readers wanting to see different treatments of the sheaf pair­ ing leading to the more sheafy Riemann-Roch for admissible metrics can also look up Moret-Bailly [MB], who assumes semistability (but a proof can be given without, as in the present book). Cf. Remark 3 at the end of Chapter V, §3. In addition, the formulation of Riemann -Roch as a metric isomorph­ ism of line sheaves should be carried out over the moduli space, as already noted by Faltings [Fa 1] when he states, p. 415: "Of course our Riemann-Roch theorem applies only to a single curve and not to a family. But the technique of the proof can be generalized." Faltings also points out how such a generalized version has applications to a refined study of the calculus on arithmetic surfaces, involving the Noether for­ mula and beyond. At the moment, 1 did not find these topics ripe enough for me to expand the present book to include them. For one thing, appropriate basic texts on the moduli scheme are not yet available for reference, e.g. Chai-Faltings, [A-C-G-H] Volume II, etc. However, 1 thank Vojta for providing me with an appendix which explains how Arakelov theory, suitably developed, implies certain diophantine inequali­ ties, and more generally how Vojta's basic diophantine conjectures are related to bounds for the canonical sheaf (so-called dualizing sheaf). The level of exposition of this appendix is considerably higher than for the rest of the book, but an attempt has been made to make it as accessible as possible by repeating some definitions and giving some discussion of concepts involved. Readers should also be aware of the development of the higher dimensional intersection theory by Gillet-Soule, and the ana­ lytic contributions of Quillen, Bost, and Bismut, in course of publication. I have tried to make the exposition as self-contained as possible, refer­ ring to books and systematic expositions of more elementary material whenever necessary. For the Neron-Tate form, the local Neron symbols, and the construction of the Neron functions (metrics on line bundles) on curves, as pull backs from the Jacobian via theta functions, I could refer to my Fundamentals of Diophantine Geometry. For various topics in analysis, I was able to refer to appropriate texts, for instance, Griffiths­ Harris to construct Green's functions (solving a aJ-equation), and to other books for other facts about PDEs. As for algebraic geometry, I give a proof of the residue theorem due to Kunz-Waldi, based on Kunz's book Kiihler Differentials, which is part of basic algebraic geo­ metry, and allows one to see directly the algebraic part of the adjunction formula in the arithmetic case. I find this extremely satisfactory. It would also be nice if eventually expositions of residues and the dualizing sheaf in the case of local complete intersections gave a proper account of this theorem, which can also be viewed as imbedded in the Grothendieck theory, but it turns out that duality theory was not needed here. The situation with respect to semi stability is much more disagreeable. I have listed the results required for Faltings' theorem in Chapter V, §5, FOREWORD Vll and I have included those proofs for which I could find no convenient reference. I am indebted to Mike Artin for his guidance to the literature (especially some of his early papers in this respect), as well as for other useful suggestions.
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