POeip-Cheur Hautor Theory Chung-Chun Yang and Indefi nite Inner Product Spaces Presented on the occasion of the retirement of Heinz Langer in the Colloquium on Operator Theory, Vienna, March 2004 Matthias Langer Annemarie Luger Harald Woracek Editors Birkhäuser Verlag . . Basel Boston Berlin AEduitthoorrsss::: MPeai-ttChhiaus HLuanger Chung-ChAunn nYeamngarie Luger DDeeppaarrttmmeenntt ooff MMaatthheemmaattiiccss DepartmenHt aorfa lMd aWthoermacaetikcs Shandong University The Hong Kong University of Science and Technology University of Strathclyde Institut für Analysis und Scienti(cid:1) c Computing Jinan 250100, Shandong Clear Water Bay Road 26 Richmond Street Technische Universität Wien P.R. China Kowloon, Hong Kong Glasgow G1 1XH Wiedner Hauptstrasse 8–10 / 101 e-mail: [email protected] P.R. China U K e-mail: ma1y0a4n0g @Wuiesnt.hk e-mail: [email protected] Austria e-mail: [email protected] [email protected] 2000 Mathematics Subject Classi(cid:1) cation Primary 46C20, 47B50; Secondary 34L05, 47A57, 47A75 2000 Mathematics Subject Classifi cation 11Dxx, 11E95, 11F03, 11G05, 11Mxx, 11P05; 30D35, 30G06; 32A22 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogra(cid:1) e; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-77556185--X9 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speci(cid:1) cally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on micro(cid:1) lms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF (cid:1) Cover design: MHeicinhza HLiolttrborvusnknye,r ,C BHa-s4e1l06 Therwil, Switzerland Printed in Germany ISBN-10: 3-7643-77551658--9X e-ISBN: 3-7643-77556196--87 ISBN-13: 978-3-7643-77556185--37 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Heights 1.1 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fields with valuations . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Extensions of absolute values . . . . . . . . . . . . . . . . . 16 1.3 Discriminant of field extensions . . . . . . . . . . . . . . . . . . . . 21 1.3.1 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.2 Dedekind domain . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.3 Different . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 Product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5 Hermitian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.1 Gauges of elementary operators . . . . . . . . . . . . . . . . 36 1.5.2 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.5.3 Nochka weights . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.6 Basic geometric notions . . . . . . . . . . . . . . . . . . . . . . . . 50 1.6.1 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.6.2 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.6.3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.6.4 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . 57 1.6.5 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.6.6 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.7 Weil functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.8 Heights in number fields . . . . . . . . . . . . . . . . . . . . . . . . 73 1.9 Functorial properties of heights . . . . . . . . . . . . . . . . . . . . 78 1.10 Gauss’ lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 vi Contents 2 Nevanlinna Theory 2.1 Notions in complex geometry . . . . . . . . . . . . . . . . . . . . . 89 2.1.1 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . 89 2.1.2 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . 93 2.1.3 Meromorphic mappings . . . . . . . . . . . . . . . . . . . . 103 2.1.4 Holomorphic vector bundles . . . . . . . . . . . . . . . . . . 106 2.1.5 Holomorphic line bundles . . . . . . . . . . . . . . . . . . . 116 2.1.6 Hermitian manifolds . . . . . . . . . . . . . . . . . . . . . . 121 2.1.7 Parabolicmanifolds . . . . . . . . . . . . . . . . . . . . . . 126 2.1.8 Inequalities on symmetric polynomials . . . . . . . . . . . . 128 2.2 Kobayashihyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . 130 2.2.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . 130 2.2.2 Measure hyperbolicity . . . . . . . . . . . . . . . . . . . . . 132 2.2.3 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.3 Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . 139 2.4 Growth of rational functions. . . . . . . . . . . . . . . . . . . . . . 146 2.5 Lemma of the logarithmic derivative . . . . . . . . . . . . . . . . . 150 2.6 Second main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 157 2.7 Notes on the second main theorem . . . . . . . . . . . . . . . . . . 163 2.8 The Cartan-Nochka theorem . . . . . . . . . . . . . . . . . . . . . 167 2.9 First main theorem for line bundles . . . . . . . . . . . . . . . . . . 175 2.10 Jacobian sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 2.11 Stoll’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.12 Carlson-Griffiths-King theory . . . . . . . . . . . . . . . . . . . . . 197 2.12.1 Second main theorem for line bundles . . . . . . . . . . . . 197 2.12.2 Griffiths’ and Lang’s conjectures . . . . . . . . . . . . . . . 205 3 Topics in Number Theory 3.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.1.1 The geometry of elliptic curves . . . . . . . . . . . . . . . . 213 3.1.2 Modular functions . . . . . . . . . . . . . . . . . . . . . . . 218 3.1.3 Cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 3.1.4 Problems in elliptic curves . . . . . . . . . . . . . . . . . . . 226 3.1.5 Hyperelliptic and rational curves . . . . . . . . . . . . . . . 231 3.2 The abc-conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 3.3 Mordell’s conjecture and generalizations . . . . . . . . . . . . . . . 237 3.4 Fermat equations and Waring’s problem . . . . . . . . . . . . . . . 240 3.5 Thue-Siegel-Roth’s theorem . . . . . . . . . . . . . . . . . . . . . . 243 3.6 Schmidt’s subspace theorem . . . . . . . . . . . . . . . . . . . . . . 245 3.7 Vojta’s conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Contents vii 3.8 Subspace theorems on hypersurfaces . . . . . . . . . . . . . . . . . 256 3.8.1 Main results and problems. . . . . . . . . . . . . . . . . . . 256 3.8.2 Equivalence of Theorems 3.65 and 3.68. . . . . . . . . . . . 260 3.8.3 Proof of Theorem 3.65 . . . . . . . . . . . . . . . . . . . . . 263 3.8.4 Proof of Theorem 2.63 . . . . . . . . . . . . . . . . . . . . . 270 3.9 Vanishing sums in function fields . . . . . . . . . . . . . . . . . . . 272 3.9.1 Algebraic function fields . . . . . . . . . . . . . . . . . . . . 272 3.9.2 Mason’s inequality . . . . . . . . . . . . . . . . . . . . . . . 275 3.9.3 No vanishing subsums . . . . . . . . . . . . . . . . . . . . . 278 4 Function Solutions of Diophantine Equations 4.1 Nevanlinna’s third main theorem . . . . . . . . . . . . . . . . . . . 287 4.2 Generalized Mason’s theorem . . . . . . . . . . . . . . . . . . . . . 296 4.3 Generalized abc-conjecture . . . . . . . . . . . . . . . . . . . . . . . 301 4.4 Generalized Hall’s conjecture . . . . . . . . . . . . . . . . . . . . . 304 4.5 Borel’s theorem and its analogues . . . . . . . . . . . . . . . . . . . 308 4.5.1 Borel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 308 4.5.2 Siu-Yeung’s theorem . . . . . . . . . . . . . . . . . . . . . . 315 4.5.3 Analogue of Borel’s theorem. . . . . . . . . . . . . . . . . . 317 4.6 Meromorphic solutions of Fermat equations . . . . . . . . . . . . . 319 4.7 Waring’s problem for meromorphic functions . . . . . . . . . . . . 328 4.8 Holomorphic curves into a complex torus . . . . . . . . . . . . . . 336 4.8.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 337 4.8.2 Complex torus . . . . . . . . . . . . . . . . . . . . . . . . . 339 4.9 Hyperbolic spaces of lower dimensions . . . . . . . . . . . . . . . . 341 4.9.1 Picard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 341 4.9.2 Hyperbolic curves . . . . . . . . . . . . . . . . . . . . . . . 343 4.9.3 Hyperbolic surfaces. . . . . . . . . . . . . . . . . . . . . . . 350 4.9.4 Uniqueness polynomials . . . . . . . . . . . . . . . . . . . . 355 4.10 Factorization of functions . . . . . . . . . . . . . . . . . . . . . . . 356 4.11 Wiman-Valiron theory . . . . . . . . . . . . . . . . . . . . . . . . . 364 5 Functions over Non-Archimedean Fields 5.1 Equidistribution formula . . . . . . . . . . . . . . . . . . . . . . . . 371 5.2 Second main theorem of meromorphic functions . . . . . . . . . . . 379 5.3 Equidistribution formula for hyperplanes . . . . . . . . . . . . . . . 384 5.4 Non-Archimedean Cartan-Nochka theorem. . . . . . . . . . . . . . 389 5.5 Holomorphic curves into projective varieties . . . . . . . . . . . . . 394 5.5.1 Equidistribution formula for hypersurfaces. . . . . . . . . . 394 5.5.2 Characteristic functions for divisors . . . . . . . . . . . . . 397 viii Contents 5.6 The abc-theorem for meromorphic functions . . . . . . . . . . . . . 399 5.7 The abc-theorem for entire functions . . . . . . . . . . . . . . . . . 403 5.8 Non-Archimedean Borel theorem . . . . . . . . . . . . . . . . . . . 406 5.9 Waring’s problem over function fields . . . . . . . . . . . . . . . . . 410 5.10 Picard-Berkovich’stheorem . . . . . . . . . . . . . . . . . . . . . . 413 6 Holomorphic Curves in Canonical Varieties 6.1 Variations of the first main theorem . . . . . . . . . . . . . . . . . 421 6.1.1 Green’s formula. . . . . . . . . . . . . . . . . . . . . . . . . 421 6.1.2 Analogue of Vojta’s conjecture . . . . . . . . . . . . . . . . 423 6.2 Meromorphic connections . . . . . . . . . . . . . . . . . . . . . . . 429 6.3 Siu theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 6.3.1 Siu’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . 435 6.3.2 Generalization of Siu’s theorem . . . . . . . . . . . . . . . . 441 6.4 Bloch-Green’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . 444 6.5 Green-Griffiths’ conjecture . . . . . . . . . . . . . . . . . . . . . . . 447 6.6 Notes on Griffiths’ and Lang’s conjectures . . . . . . . . . . . . . . 450 7 Riemann’s ζ-function 7.1 Riemann’s functional equation . . . . . . . . . . . . . . . . . . . . 461 7.2 Converse theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 7.3 Riemann’s hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 473 7.4 Hadamard’s factorization . . . . . . . . . . . . . . . . . . . . . . . 480 7.5 Nevanlinna’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 486 7.6 Carleman’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 7.7 Levin’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 7.8 Notes on Nevanlinna’s conjecture . . . . . . . . . . . . . . . . . . . 504 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Preface In 1879, Picard established the well-known and beautiful result that a transcen- dentalentire function assumesallvaluesinfinitely oftenwith one exception.Since then Hadamard(1893),Borel(1897)and Blumenthal (1910)had tried to give Pi- card’s result a quantitative description and extend it to meromorphic functions. ItwasR.Nevanlinna,whoachievedsuchanattemptin(1925)by establishingthe so-called value distribution theory of meromorphic functions which was praised by H. Weyl (1943)as “One of the few greatmathematical events of our century”. Moreover, part of the significance of Nevanlinna’s approach is that the concept of exceptional values can be given a geometric interpretation in terms of geomet- ric objects like curves and mappings of subspaces of holomorphic curves from a complex plane C to a projective space Pn. In the years since these results were achieved, mathematicians of comparable stature have made efforts to derive an analogous theory for meromorphic mappings and p-adic meromorphic functions. Besides the value distribution, the theory has had many applications to the an- alyticity, growth, existence, and unicity properties of meromorphic solutions to differential or functional equations. More recently, it has been found that there is a profound relation between Nevanlinna theory and number theory. C.F. Osgood [310], [311] first noticed a similarity between the 2 in Nevanlinna’s defect relation andthe 2 in Roth’s theorem. S. Lang [230] pointed to the existence of a structure totheerrorterminNevanlinna’ssecondmaintheorem,conjecturedwhatcouldbe essentially the best possible form of this error term in general (based on his con- jecture onthe errortermin Roth’s theorem),andgavea quitedetailed discussion in [235]. P.M. Wong [433] used a method of Ahlfors to prove Lang’s conjecture in the one-dimensional case. As for higher dimensions, this problem was studied by S. Lang and W. Cherry [235], A. Hinkkanen [159], and was finally completed by Z. Ye [443]. The best possible form of error terms has been used in our present work to produce some sharp results. In 1987, P. Vojta [415] gave a much deeper analysis of the situation, and comparedthetheoryofheightsinnumbertheorywiththecharacteristicfunctions of Nevanlinna theory. In his dictionary, the second main theorem, due to H. Car- tan,correspondstoSchmidt’ssubspacetheorem.Further,heproposedthegeneral conjecture in number theory by comparing the second main theorem in Carlson- Griffiths-King’s theory, or particularly influenced by Griffiths’ conjecture, which x Preface also can be translated into a problem of non-Archimedean holomorphic curves posed by Hu and Yang [176]. Along this route, Shiffman’s conjecture on hyper- surfacetargetsinvalue distributiontheorycorrespondsto asubspacetheoremfor homogeneouspolynomialformsinDiophantineapproximation.Vojta’s(1,1)-form conjecture is an analogue of an inequality of characteristic functions of holomor- phic curves for line bundles. Being influenced by Mason’s theorem, Oesterl´e and Masserformulatedtheabc-conjecture.Thegeneralizedabc-conjecturesforintegers are counterparts of Nevanlinna’s third main theorem and its variations in value distribution theory, and so on. Roughlyspeaking,asignificantanalogybetweenNevanlinnatheoryandDio- phantine approximation seems to be that the sets X(κ) of κ-rational points of a projective variety X defined over number fields κ are finite if and only if there are no non-constant holomorphic curves into X. Mordell’s conjecture (Faltings’ theorem) and Picard’s theorem are classic examples in this direction. In higher- dimensional spaces, this corresponds to a conjecture due to S. Lang, that is, Kobayashihyperbolic manifolds (which do not contain non-constantholomorphic curves) are Mordellic. Bloch-Green-Griffiths’ conjecture on degeneracy of holo- morphic curves into pseudo-canonical projective varieties is an analogue of the Bombieri-Langconjectureonpseudocanonicalvarieties.Wehaveintroducedthese problemsandtherelateddevelopmentsinthisbook.Generally,topicsorproblems in number theory are briefly introduced and translated as analogues of topics in value distributiontheory. We haveomitted the proofs oftheorems in number the- ory. However, we have discussed the problems of value distribution in detail. In this book, we will not discuss value distribution theory of moving targets, say, K. Yamanoi’s work [437], and their counterparts in number theory. When a holomorphic curve f into X is not constant, we have to distinguish whether it is degenerate in Nevanlinna theory, that is, whether its image is con- tained or not in a proper subvariety. If it is degenerate, usually it is difficult to deal with it in value distribution theory. If f is non-degenerate, we can study its valuedistributionsandmeasureitsgrowthwellbyacharacteristicfunctionT (r). f Similarly, we should distinguish whether or not certain rationalpoints are degen- erate. Related to the degeneracy, it seems that for each number field κ, X(κ) is containedinaproperZariskiclosedsubsetifandonlyiftherearenoalgebraically non-degenerate holomorphic curves into X. To compare with Nevanlinna theory, therefore, we need to rule out degenerate κ-rational points that are contained in a subspace of lower dimension, and give a proper measure for non-degenerate κ- rational points. By integrating heights over non-degenerate κ-rational points, we can obtain quantitative measurements T (r). κ They have the following basic properties: (i) f is constant if and only if T (r) is bounded; there are no non-degenerate f κ-rational points if and only if T (r) is bounded . κ (ii) f is rational if and only if T (r) = O(logr); there are only finitely many f non-degenerate κ-rational points if and only if T (r)=O(logr). κ Preface xi Ithasbeenobservedthatthereexistnon-constantholomorphiccurvesintoelliptic curvessuchthattheymustbesurjective.Thusitispossiblethatthereareinfinitely many rational points on some elliptic curves. However, since non-constant holo- morphiccurvesintoellipticcurveshavenormalproperties,say,theyaresurjective, then distribution of rational points on elliptic curves should be “normal”. Really, elliptic curves are modular according to the Shimura-Taniyama-Weil conjecture, whichwasprovedbyBreuil,Conrad,Diamond,andTaylor[37]byextendingwork ofWiles[431],TaylorandWiles[390].Moreover,asaresultofstudiesofthe anal- ogousresultsbetweenNevanlinna’svaluedistributiontheoryandDiophantineap- proximation,somenovelideasandgeneralizationshavebeendevelopedorderived in the two topics, with many open problems posed for further investigations. The book consists of seven chapters: In Chapter 1, we introduce some basic notationandterminology onfields andalgebraicgeometry which aremainly used to explain clearly the topics in Chapter 3 related to number theory. Chapter 2 is a foundation of value distribution theory which is used in Chapter 4, Chapter 6 andChapter7tointroducetheanaloguesrelatedtonumbertheoryinNevanlinna theory, say, abc-problems, meromorphic solutions of Fermat’s equations and the Waring problem, Green-Griffiths’ conjecture, Griffiths’ and Lang’s conjectures, Riemann’s ζ-function, and so on. Chapter 5 contains value distribution theory over non-Archimedean fields and some applications related to topics in number theory.Moreover,a few equidistribution formulae illustrating the differences with the classical Nevanlinna theory have been exhibited. Each chapter of this book is self-contained and this book is appended with a comprehensive and up-dated list of references. The book will provide not just some new research results and directions but challenging open problems in studying Diophantine approximation andNevanlinna theory.One ofthe aims of this book is to maketimely surveyson thesenewresultsandtheirrelateddevelopments;someofwhicharenewlyobtained by the authors and have not been published yet. It is hoped that the publication of this book will stimulate, among our peers, further researches on Nevanlinna’s value distribution theory, Diophantine approximation and their applications. Wegratefullyacknowledgesupportfortherelatedresearchandforwritingof thepresentbookfromtheNaturalScienceFundofChina(NSFC)andtheResearch Grant Council of Hong Kong during recent years. Also the authors would like to thank the staff of Birkha¨user, in particular, the Head of Editorial Department STM, Dr. Thomas Hempfling, and last but not least, we want to express our thanks to Dr. Michiel Van Frankenhuijsen for his thorough reviewing, valuable criticism and concrete suggestions. Pei-Chu Hu Chung-Chun Yang