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TranscendenceandLinearRelationsof1-Periods.ByA.HuberandG.Wu¨stholz The Mordell Conjecture A Complete Proof from Diophantine Geometry HIDEAKI IKOMA ShitennojiUniversity SHU KAWAGUCHI DoshishaUniversity ATSUSHI MORIWAKI KyotoUniversity UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314-321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi-110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108845953 DOI:10.1017/9781108991445 ©HideakiIkoma,ShuKawaguchiandAtsushiMoriwaki2022 Mordell-FaltingsNOTEIRIbyHideakiIkoma,ShuKawaguchiandAtsushiMoriwaki. OriginalJapaneselanguageeditionpublishedbySaiensu-shaCo.,Ltd. 1-3-25,Sendagaya,Shibuya-ku,Tokyo151-0051,JapanCopyright©2017, Saiensu-shaCo.,Ltd.AllRightsreserved. Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-84595-3Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Preface pagevii 1 WhatIstheMordellConjecture(Faltings’sTheorem)? 1 2 SomeBasicsofAlgebraicNumberTheory 6 2.1 TraceandNorm 6 2.2 AlgebraicIntegersandDiscriminants 9 2.3 IdealsintheRingofIntegers 11 2.4 LatticesandMinkowski’sConvexBodyTheorem 14 2.5 Minkowski’sDiscriminantTheorem 17 2.6 FieldExtensionandRamificationIndex 22 3 TheoryofHeights 25 3.1 AbsoluteValues 25 3.2 ProductFormula 27 3.3 HeightsofVectorsandPointsinProjectiveSpace 29 3.4 HeightFunctionsAssociatedtoLineBundles 33 3.5 Northcott’sFinitenessTheorem 39 3.6 IntroductiontoAbelianVarieties 43 3.7 HeightFunctionsonAbelianVarieties 52 3.8 CurvesandTheirJacobians 59 3.9 TheMordell–WeilTheorem 67 4 PreliminariesfortheProofofFaltings’sTheorem 73 4.1 Siegel’sLemma 73 4.2 InequalitiesonLengthsandHeightsofPolynomials 77 v vi Contents 4.3 RegularLocalRingandIndex 85 4.4 Roth’sLemma 88 4.5 NormsofInvertibleSheaves 95 4.6 HeightofNorm 99 4.7 LocalEisensteinTheorem 111 5 TheProofofFaltings’sTheorem 117 5.1 KeysfortheProofofFaltings’sTheorem 117 5.2 TechnicalSettingsfortheProofsofTheorem5.4,Theorem5.5, andTheorem5.6 129 5.3 ExistenceofSmallSection(theProofofTheorem5.4) 134 5.4 UpperBoundoftheIndex(theProofofTheorem5.5) 143 5.5 LowerBoundoftheIndex(theProofofTheorem5.6) 146 5.6 AnApplicationtoFermatCurves 156 References 160 Notation 163 IndexofSymbols 166 Index 167 Preface This book originated from course notes for “Topics in Algebra” taught by Atsushi Moriwaki to senior undergraduate students and beginning graduate studentsatKyotoUniversityin1996.ShuKawaguchi,thenagraduatestudent, attendedthecourse. The purpose of the course was to give a self-contained and detailed proofoftheMordellconjecture(Faltings’stheorem)byfollowingVojta’sand Bombieri’spapers[5,29],whiletouchingonseveralimportanttheoremsand techniquesfromDiophantinegeometry. We have fully revised and expanded the course notes into this book, and some of the explicit and detailed computations presented here may be appearing in the literature for the first time. This book will also provide an introductiontoDiophantinegeometry. We assume that the reader is familiar with basic concepts of algebraic geometry and has good knowledge of undergraduate algebra and analysis. SomebasicsofalgebraicnumbertheoryareincludedinChapter2. ForthereaderwhoisfamiliarwiththebasicsofDiophantinegeometry,and isinterestedonlyintheproofofFaltings’stheorem,wesuggeststartingfrom Chapter 5 while referring to Chapter 4. Otherwise, we suggest starting with Chapters2and3whilereferring,ifnecessary,tobooksonalgebraicgeometry andalgebraicnumbertheory(e.g.,[11,23]),andthenreadingChapter5while referringtoChapter4. vii 1 What Is the Mordell Conjecture (Faltings’s Theorem)? Diophantine geometry is the field of mathematics that concerns integer solutions and rational solutions of polynomial equations. It is named after Diophantus of Alexandria from around the third century who wrote a series ofbookscalledArithmetica.Diophantinegeometryisoneoftheoldestfields of mathematics, and it continues to be a major field in number theory and arithmetic geometry. If integer solutions and rational solutions are put aside, then polynomial equations determine an algebraic variety. Since around the start of the twentieth century, algebro-geometric methods have played an importantroleinthestudyofDiophantinegeometry. In 1922, Mordell (Figure 1.1) made a surprising conjecture in a paper where he proved the so-called Mordell–Weil theorem for elliptic curves (see Theorem 3.42). This conjecture, called the Mordell conjecture before Faltings’s proof appeared, states that the number of rational points is finite on any geometrically irreducible algebraic curve of genus at least 2 defined over a number field. It is not certain on what grounds Mordell made this conjecture, but it was audacious at the time, and attracted the attention of manymathematicians.Whilesomepartialresultswereobtained,theMordell conjecturestoodasanunclimbedmountainbeforetheproofbyFaltings.Thus, whenFaltings(Figure1.2)provedtheMordellconjectureinapaperpublished in 1983, the news was circulated around the globe with much enthusiasm. Faltings’sproofwasmomentous,usingsophisticatedandprofoundtheoriesof arithmetic geometry. He proved the Shafarevich conjecture, the Tate conjec- ture,andtheMordellconjectureconcurrently,andhewasawardedtheFields Medalin1986.Nevertheless,first-yearstudentsatuniversitiescanunderstand thestatementoftheMordellconjecture,exceptforthenotionofgenus. Let f(X,Y) be a two-variable polynomial with coefficients in a number fieldK (e.g.,thefieldQofrationalnumbers).Weassumethefollowing: 1 2 WhatIstheMordellConjecture? Figure1.1 LouisJ.Mordell. Source:ArchivesoftheMathematischesForschungsinstitutOberwolfach. Figure1.2 GerdFaltings. Source:ArchivesoftheMathematischesForschungsinstitutOberwolfach. 1. f(X,Y)isirreducibleasapolynomialinC[X,Y].Namely,if f(X,Y)=g(X,Y)h(X,Y)withg(X,Y),h(X,Y)∈C[X,Y],theng(X,Y) orh(X,Y)isaconstant. 2. ThealgebraiccurveC definedbyf(X,Y)=0,extendedtotheprojective plane,issmooth.Inotherwords,letF(X,Y,Z)∈C[X,Y,Z]bethe homogeneouspolynomialwith F(X,Y,1)=f(X,Y) and degF(X,Y,Z)=degf(X,Y). ThentheonlysolutioninC3of F(X,Y,Z)=(∂F/∂X)(X,Y,Z)=(∂F/∂Y)(X,Y,Z) =(∂F/∂Z)(X,Y,Z)=0 is(0,0,0). Inthissetting,thealgebraiccurveC hasgenusatleast2ifandonlyifthe degreeoff isatleast4.Thus,theMordellconjecturestatesthatifthedegree of f is at least 4, then the number of points (a,b) ∈ K2 with f(a,b) = 0 is finite. Here, the assumption that f is irreducible is essential. Indeed, for