A panorama of number theory, or, The view from Baker's garden PDF
Preview A panorama of number theory, or, The view from Baker's garden
This page intentionally left blank APanoramainNumberTheory or TheViewfromBaker’sGarden A Panorama in Number Theory or The View from Baker’s Garden editedby GisbertWu¨stholz ETH,Zu¨rich Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521807999 © Cambridge University Press 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 - isbn-13 978-0-511-06390-9 eBook (NetLibrary) - isbn-10 0-511-06390-3 eBook (NetLibrary) - isbn-13 978-0-521-80799-9 hardback - isbn-10 0-521-80799-9 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Contributors pagevii Introduction xi 1 OneCenturyofLogarithmicForms G.Wu¨stholz 1 2 Reporton p-adicLogarithmicForms KunruiYu 11 3 RecentProgressonLinearFormsinEllipticLogarithms Sinnou David&NorikoHirata-Kohno 26 4 SolvingDiophantineEquationsbyBaker’sTheory Ka´lma´nGyo˝ry 38 5 Baker’sMethodandModularCurves YuriF.Bilu 73 6 ApplicationoftheAndre´–OortConjecturetosomeQuestionsin Transcendence PaulaB.Cohen&GisbertWu¨stholz 89 7 RegularDessins,EndomorphismsofJacobians,andTranscendence Ju¨rgenWolfart 107 8 MaassCuspFormswithIntegerCoefficients PeterSarnak 121 9 ModularForms,EllipticCurvesandthe ABC-Conjecture Dorian Goldfeld 128 10 OntheAlgebraicIndependenceofNumbers Yu.V.Nesterenko 148 11 IdealLattices EvaBayer-Fluckiger 168 12 IntegralPointsandMordell–WeilLattices TetsujiShioda 185 13 FortyYearsofEffectiveResultsinDiophantineTheory Enrico Bombieri 194 14 PointsonSubvarietiesofTori Jan-HendrikEvertse 214 15 ANewApplicationofDiophantineApproximations G.Faltings 231 16 SearchBoundsforDiophantineEquations D.W.Masser 247 17 RegularSystems,UbiquityandDiophantineApproximation V.V.Beresnevich,V.I.Bernik&M.M.Dodson 260 18 DiophantineApproximation,LatticesandFlowsonHomogeneous Spaces GregoryMargulis 280 v vi Contents 19 OnLinearTernaryEquationswithPrimeVariables–Baker’sConstant andVinogradov’sBound Ming-ChitLiu&TianzeWang 311 20 PowersinArithmeticProgression T.N.Shorey 325 21 On the Greatest Common Divisor of Two Univariate Polynomials, I A.Schinzel 337 22 Heilbronn’sExponentialSumandTranscendenceTheory D.R.Heath-Brown 353 Contributors Eva Bayer-Fluckiger, Departement de Mathe´matiques Ecole Polytechnique FederaledeLausanne1015LausanneSwitzerland [email protected] V.V. Beresnevich, Institute of Mathematics of the Belarus Academy of Sci- ences,220072,Surganova11,Minsk,Belarus [email protected] V.I. Bernik, Institute of Mathematics of the Belarus Academy of Sciences, 220072,Surganova11,Minsk,Belarus [email protected] Yuri F. Bilu, A2X Universite´ Bordeaux I, 351, cours de la Liberation, 33405 Talence,France [email protected] Enrico Bombieri, School of Mathematics, Institute for Advanced Study, Ein- steinDrive,PrincetonNJ08540,USA [email protected] PaulaB.Cohen,MRAGATauCNRS,UFRdeMathe´matiques,BaˆtimentM2, Universite´ desSciencesetTechnologiesdeLille,59655Villeneuved’Ascq cedex,France [email protected] SinnouDavid,Universite´ P.etM.Curie(ParisVI),InstitutMathe´matiquede Jussieu, Proble`mes Diophantiens, Case 247, 4, Place Jussieu, 75252 Paris CEDEX05,France [email protected] M.M. Dodson, Department of Mathematics, University of York, York YO10 5DD,UK [email protected] vii viii Contributors Jan–Hendrik Evertse, Universiteit Leiden, Mathematisch Instituut, Postbus 9512,2300RALeiden,TheNetherlands [email protected] G. Faltings, Max-Planck-Institut fu¨r Mathematik, Vivatgasse 7, 53111 Bonn, Germany [email protected] Dorian Goldfeld, Columbia University, Department of Mathematics, New York,NY10027,USA [email protected] Ka´lma´nGyo˝ry,InstituteofMathematicsandInformatics,UniversityofDebre- cen,H-4010Debrecen,P.O.Box12,Hungary [email protected] D.R.Heath-BrownMathematicalInstitute,Oxford,UK [email protected] Noriko Hirata-Kohno, Department of Mathematics, College of Science and Technology, Nihon University, Suruga-Dai, Kanda, Chiyoda, Tokyo 101- 8308,Japan [email protected] Ming-Chit Liu, Department of Mathematics, The University of Hong Kong, Pokfulam,HongKong;andPOBox625,Alhambra,CaliforniaCA91802, USA [email protected] G. Margulis, Yale University, Department of Mathematics, PO Box 208283, NewHavenCT06520-8283,USA [email protected] D.W. Masser, Mathematisches Institut, Universita¨t Basel, Rheinsprung 21, 4051Basel,Switzerland [email protected] Yu.Nesterenko,FacultyofMechanicsandMathematics,MainBuilding,MSU, VorobjovyGory,Moscow,119899,Russia [email protected] Peter Sarnak, Princeton University, Department of Mathematics, Fine Hall, PrincetonNJ08544-1000,USA [email protected] Andrzej Schinzel, Institute of Mathematics, Polish Academy of Sciences, ul. S´niadeckich8,POBox137,00-950Warszawa,Poland [email protected]
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