EuropeanJournalofMathematics(2015)1:405–440 DOI10.1007/s40879-015-0066-0 REVIEW ARTICLE Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki IvanFesenko1 Received:21February2015/Revised:15June2015/Accepted:22June2015/ Publishedonline:8August2015 ©SpringerInternationalPublishingAG2015 FromtheEditor-in-Chief The article below by Ivan Fesenko is an introduction to inter-universalTeichmüllertheory,asdevelopedbyShinichiMochizuki,anditsappli- cationtofamousconjecturesinDiophantinegeometry.Inaseriesofpreprintsreleased severalyearsago,Mochizukiintroducedavastcollectionofnewideasandmethods relatedtoarithmeticdeformation.Thelevelofnoveltyofthetheoryhasmadeitchal- lengingtostudyevenformanyexpertsinarithmeticgeometry.Thearticlesurveysthe mainfeaturesandobjectsoftheworkofMochizukiandalsoprovidesanindependent perspective.Wehopethatthearticlewillbeofhelpinunderstandingthemainconcepts andinnovationsofthisimportanttheory. Abstract These notes survey the main ideas, concepts and objects of the work byShinichiMochizukioninter-universalTeichmüllertheory(Inter-universalTeich- müller theory I–IV, 2012–2015) which might also be called arithmetic deformation theory,anditsapplicationtodiophantinegeometry.Theyprovideanexternalperspec- tivewhichcomplementsthereviewtextsofMochizuki(Invitationtointer-universal Teichmüllertheory(lecturenoteversion),2015)and(AlgebraicNumberTheoryand RelatedTopics2012.RIMSKôkyûrokuBessatsu,volB51,pp.301–346,2014).Some important developments which preceded (Inter-universal Teichmüller theory I–IV, 2012–2015)arepresentedinthefirstsection.Severalimportantaspectsofarithmetic deformation theory are discussed in the second section. Its main theorem gives an inequality–boundonthesizeofvolumedeformationassociatedtoacertainlog-theta- lattice.Theapplicationtoseveralfundamentalconjecturesinnumbertheoryfollows from a further direct computation of the right hand side of the inequality. The third B IvanFesenko [email protected] 1 SchoolofMathematical,SciencesUniversityofNottingham,UniversityPark, Nottingham,NG72RD,UK 123 406 IvanFesenko sectionconsidersadditionalrelatedtopics,includingpracticalhintsonhowtostudy thetheory. Keywords Inter-universal Teichmüller theory · Arithmetic deformation · Key conjectures in Diophantine geometry · Fundamental groups · Mono-anabelian geometry·Nonarchimedeanthetafunctionanditsspecialvalues·Deconstruction andReconstructionofringstructures·Theta-links·Log-theta-lattice MathematicsSubjectClassification 11G99·11D99·14G99 Contents 1 Theorigins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 1.1 FromclassfieldtheorytoreconstructingnumberfieldstocoveringsofP1minusthreepoints 407 1.2 Adevelopmentindiophantinegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 1.3 Conjecturalinequalitiesforthesameproperty . . . . . . . . . . . . . . . . . . . . . . . . 409 1.4 Aquestionposedtoastudentbyhisthesisadvisor . . . . . . . . . . . . . . . . . . . . . . 412 1.5 Onanabeliangeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 2 Onarithmeticdeformationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 2.1 TextsrelatedtoIUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 2.2 Initialdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 2.3 Abriefoutlineoftheproofandalistofsomeofthemainconcepts . . . . . . . . . . . . . 416 2.4 Mono-anabeliangeometryandmultiradiality . . . . . . . . . . . . . . . . . . . . . . . . . 419 2.5 Nonarchimedeantheta-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 2.6 GeneralisedKummertheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 2.7 Thetheta-linkandtwotypesofsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 423 2.8 Nonarchimedeanlogarithm,log-link,log-theta-lattice,log-shell . . . . . . . . . . . . . . . 426 2.9 Rigiditiesandindeterminacies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 2.10 Theroleofglobaldata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 2.11 ThemaintheoremofIUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 2.12 TheapplicationofIUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 2.13 Moretheorems,objectsandconceptsofIUT . . . . . . . . . . . . . . . . . . . . . . . . . 432 2.14 AnalogiesandrelationsbetweenIUTandothertheories . . . . . . . . . . . . . . . . . . . 432 3 StudyingIUTandrelatedaspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 3.1 OntheverificationofIUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 3.2 EntrancestoIUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 3.3 TheworkofShinichiMochizuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 3.4 Relatedissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Foreword.Theaimofthesenotesistopresent,inarelativelysimpleform,thekey ideas, concepts and objects of the work of Shinichi Mochizuki on inter-universal Teichmüllertheory(IUT),toasmanypotentialreadersaspossible.Thepresentation isbasedonmyownexperienceinstudyingIUT.Thistextisexpectedtohelpitsreaders togainageneraloverviewofthetheoryandacertainorientationinit,aswellasto seevariouslinksbetweenitandexistingtheories. ReadingthesenotescannotreplaceorsubstituteaseriousstudyofIUT.Asmen- tioned in [42], there are currently no shortcuts in the study of IUT. Hence there are probablytwomainoptionsavailableatthetimeofwritingofthistexttolearnabout 123 NotesonthetheoryofShinichiMochizuki 407 theessenceofIUT.Thefirstistodedicateasignificantamountoftime1tostudyingthe theorypatientlyandgraduallyreachingitsmainparts.IrefertoSect.3formypersonal adviceonhowtostudytheoriginaltextsofarithmeticdeformationtheory,whichis anothername(duetotheauthorofthepresenttext)forthetheory.Thesecondoption istoreadthereviewtexts[39,40]andintroductionsofpapers,etc.Myexperienceand the experience of several other mathematicians show that the review texts could be hard tofollow, and reading them before a serious study of IUT may be not the best way, while reading them after some preliminary study or in the middle of it can be moreuseful.Weshallseetowhatextentthisfeatureissharedbythesenotes. In view of the declared aim of this text and the natural limitation on its size, it is inevitable that several important mathematical objects and notions in Sect. 2 are introducedinavagueform.OneofthemostimportantnovelobjectsofIUTisaso- calledtheta-link.Averylargepartof[35,37]definesthetheta-linkanddevelopsits enhancedversions.2 1 Theorigins 1.1 FromclassfieldtheorytoreconstructingnumberfieldstocoveringsofP1 minusthreepoints Abelian class field theory for one-dimensional global and local fields, in particular for number fields and their completions, has played a central role in number theory andstimulatedmanyfurtherdevelopments.InverseGaloistheory,severalversionsof theLanglandsprogramme,anabeliangeometry,(abelian)higherclassfieldtheoryand higher adelic geometry and analysis, and, to some extent, the arithmetic of abelian varietiesoverglobalfieldsandtheircompletionsareamongthem. InverseGaloistheorystudieshowtorealisefiniteorinfinitecompacttopological groupsasGaloisgroupsofvariousfieldsincludingextensionsofnumberfieldsand their completions, see e.g. [21]. For abelian groups and local and global fields, the answer follows from class field theory. A theorem of Shafarevich states that every soluble group can be realised as a Galois group over a global field, see e.g. [47, Section6,ChapterIX]. Let Kalg be an algebraic closure of a number field K. The Galois group G = K G(Kalg/K)iscalledtheabsoluteGaloisgroupof K. TheNeukirch–Ikeda–Uchidatheorem(provedbytheendof1970s;theproofused globalclassfieldtheory)asserts,seee.g.[47,Section2,ChapterXII],thefollowing: For two number fields K ,K and any isomorphism of topological groups 1 2 ψ: G −→∼ G there is a unique field isomorphism σ: Kalg −→∼ Kalg such K1 K2 2 1 that σ(K ) = K and ψ(g)(a) = σ−1(gσ(a)) for alla ∈ Kalg, g ∈ G . In 2 1 2 K1 1 Inmyopinion,atleast250–500h. 2 Anappreciationofthegeneralqualitativeaspectsofthetheta-linkmaybeobtainedbystudyingthe simplestversionofthetheta-link.Thisversionisdiscussedin[35,SectionI1],whiletechnicaldetails concerningtheconstructionofthisversionmaybefoundinapproximately30pagesof[35,Section3]. 123 408 IvanFesenko particular, the homomorphism GQ → Aut(GQ), g (cid:5)→ (h (cid:5)→ ghg−1), is an isomorphismandeveryautomorphismofGQisinner. Thenexttheoremwasincludedas[2,Theorem4]in1980,aspartofBelyi’sstudyof aspectsofinverseGaloistheory: AnirreduciblesmoothprojectivealgebraiccurveC definedoverafieldofchar- acteristiczerocanbedefinedoveranalgebraicclosureQalgofthefieldofrational numbers Q if and only if there is a covering C → P1 which ramifies over no morethanthreepointsofP1. This theorem3 plays a key role in the study of Galois groups, including algebraic and geometric fundamental groups of curves over number fields and local fields.4 Coverings of the type which appears in this theorem are often called Belyi maps. Various versions of Belyi maps are used in arithmetic deformation theory and its applicationtodiophantinegeometry. 1.2 Adevelopmentindiophantinegeometry In1983FaltingsprovedtheMordellconjecture,afundamentalfinitenesspropertyin diophantinegeometry[7,9].TheFaltings–Mordelltheoremassertsthat A curve C of genus > 1 defined over an algebraic number field K has only finitelymanyrationalpointsover K. Severalotherproofsfollowed.5VojtafoundinterestinglinkswithNevanlinnatheoryin complexanalysiswhichledtooneofhisproofs.Fortextbookexpositionsofsimplified proofssee[5,15]. 3 Thefirstversionof[2]andBelyi’sseminartalkinMoscowdealtwithanellipticcurve,whichwasenough foritssubsequentapplication,see1.5.Afterreadingthefirstversionof[2],Bogomolovnoticedthatthe originalproofofthetheoreminitworksforarbitrarycurves.HetoldBelyihowimportantthisextended versionisandurgedhimtoincludetheextendedversioninthepaper.Belyiwasreluctanttoincludethe extendedversion,onthegroundsthatoverfinitefieldseveryirreduciblesmoothprojectivealgebraiccurve maybeexhibitedasacoveringoftheprojectivelinewithatmostoneramificationpoint.Bogomolovthen talkedwithShafarevich,whoimmediatelyappreciatedthevalueoftheextendedversionandinsistedthatthe authorincludeitin[2].BogomolovfurtherdevelopedthetheoryofBelyimaps,inparticular,inrelationto theuseofcolouredRiemannsurfacesanddeliverednumeroustalksonthesefurtherdevelopments.Several yearslaterthistheoryappeared,independently,inGrothendieck’stext[13]. 4 Grothendieckwroteaboutthe“onlyif”part:“never,withoutadoubt,wassuchadeepanddisconcerting resultprovedinsofewlines!”[13]. 5 Hereweareinthebestpossiblesituationwhenaconjectureisstatedoveranarbitraryalgebraicnumber fieldandisestablishedoveranarbitraryalgebraicnumberfield,andthemethodsoftheproofsdonotdepend onthespecificfeaturesofthenumberfieldunderconsideration.Thisisnotsointhecaseofthearithmetic Langlandscorrespondence,evenforellipticcurvesovernumberfields.Inthehistoryofclassfieldtheory, theinitialperiodofdevelopingspecialtheoriesthatworkonlyoversmallnumberfieldswasfollowedbya phaseofgeneralfunctorialclassfieldtheoryoverarbitraryglobalandlocalfields.Thegeneraltheorywas eventuallyclarifiedandsimplified,see[46],anditbecameeasierthanthoseinitialtheories.Weareyetto witnessasimilarphasewhichinvolvesageneralfunctorialtheorythatworksoverarbitrarynumberfields inthecaseoftheLanglandsprogrammeandhence,inparticular,yieldsanotherproofoftheWiles–Fermat theoremviatheautomorphicpropertiesofellipticcurvesoveranynumberfield. 123 NotesonthetheoryofShinichiMochizuki 409 The same year Grothendieck wrote a letter [14] to Faltings, which proposed ele- ments of anabelian geometry. With hindsight, one of the issues raised in it was a generalisationoftheNeukirch–Ikeda–Uchidatheorem6toanabeliangeometricobjects suchashyperboliccurvesovernumberfieldsandpossibleapplicationsofanabelian geometrytoprovidenewproofsandstrongerversions,aswellasabetterunderstand- ing,ofsuchresultsindiophantinegeometryastheFaltings–Mordelltheorem,cf.1.5. 1.3 Conjecturalinequalitiesforthesameproperty There are several closely related conjectures, proposed in the period from 1978 to 1987,whichextendfurtherthepropertystatedintheMordellconjecture: (a) theeffectiveMordellconjecture—aconjecturalextensionoftheFaltings–Mordell theoremwhichinvolvesaneffectiveboundontheheightofrationalpointsofthe curveCoverthenumberfieldKintheFaltingstheoremintermsofdataassociated toC and K, (b) theSzpiroconjecture,seebelow, (c) theMasser–Oesterléconjecture,a.k.a.theabcconjecture(whosestatementover Qiswellknown,7andwhichhasanextensiontoarbitraryalgebraicnumberfields, see[5,Conjecture14.4.12]), (d) theFreyconjecture,see[15,ConjectureF.3.2(b)], (e) theVojtaconjectureonhyperboliccurves,seebelow, (f) arithmeticBogomolov–Miyaoka–Yauconjectures(thereareseveralversions). The Szpiro conjecture was stated several years before8 the work of Faltings, who learnedmuchaboutthesubjectrelatedtohisprooffromSzpiro.UsingtheFreycurve9, itis not difficult toshow that(c) and (d) are equivalent and thatthey imply (b),see e.g.see[15,SectionF3]andreferencestherein.UsingBelyimapsasin1.1,onecan show the equivalence of (c) and (a). For the equivalence of (c) and (e) see e.g. [5, Theorem14.4.16]and[54].Forimplications(e)⇒(f)see[55]. OverthecomplexnumbersthepropertyanalogoustotheSzpiroconjectureisvery interesting. For a smooth projective surface equipped with a structure of non-split minimalellipticsurfacefibredoverasmoothprojectiveconnectedcomplexcurveof genusg,suchthatthefibrationadmitsaglobalsection,and,moreover,everysingular fibreofthefibrationisoftypeI ,i.e.itscomponentsareprojectivelineswhichintersect n transversally and form an n-gon, this property states that the sum of the number of components of singular fibres does not exceed six times the sum of the number of 6 ItappearsthatGrothendieckwasnotawareofthistheorem. 7 Foreveryε > 0thereisaconstantsuchthatforallnon-zerointegersa,b,csuchthat(cid:2)(a,b,c) = 1 the equality a+b+c = 0 implies max(log|a|,log|b|,log|c|) (cid:2) constant+(1+ε) p|abclogp whereprunsthroughallpositiveprimesdividingabc.Whilethestatementoftheabcconjecturedoesnot revealimmediatelyanyunderlyinggeometricstructure,theotherconjecturesaremoregeometrical.Foran entertainingpresentationofaspectsoftheabcconjectureandrelatedproperties,seee.g.[56]. 8 In1978Szpirotalkedaboutitwithseveralmathematicians.Hemadetheconjecturepublicatameeting oftheGermanMathematicalSociety(DMV)in1982whereFrey,OesterléandMasserwerepresent. 9 y2=x(x+a)(x−b)wherea,b,a+barenon-zerocoprimeintegers. 123 410 IvanFesenko singularfibresandof2g−2.Thepropertyhasseveralproofs,ofwhichthefirstwas given by Szpiro. Shioda deduced the statement in two pages of computations from argumentsalreadyknowntoKodaira.Thesetwoproofsofthegeometricversionofthe inequalityusethecotangentbundleandtheKodaira–Spencermap.A(full)arithmetic version of the Kodaira–Spencer map could be quite useful for giving a proof of the arithmetic Szpiro conjecture. However, such an arithmetic version of the Kodaira– Spencermapisnotyetknown. Among several other proofs of this property, a proof by Bogomolov uses mon- odromyactionsandthehyperbolicgeometryoftheupperhalf-planeanddoesnotuse thecotangent bundle,see[1,Section5.3].Hisproofmakesessentialuseofthefact that the n-gons determined by the singular fibres may be equipped with a common orientation,likewindmillsrevolvinginsynchronyinthepresenceofwind.Synchro- nisationofdataplaysanimportantroleinarithmeticdeformationtheoryaswell,cf. 2.10.TodevelopanarithmeticanalogueofthegeometricproofofBogomolovtoapply toprovingthearithmeticSzpiroconjecture,oneneedsakindofarithmeticanalogue ofthehyperbolicgeometryoftheupperhalf-plane,andthisisinsomesenseachieved byIUT,see2.10. The conjectural (arithmetic) Szpiro inequality states in particular that if K is a number field, then for every ε > 0 there is a real c (depending on K and ε) such that for every elliptic curve E over the number field K with split multiplicative K reductionateverybadreductionvaluation,soallsingularfibresofitsminimalregular propermodelE → SpecOK areoftype In,theweightedsumofthenumbersnv of componentsofsingularfibressatisfies (cid:3) (cid:3) nvlog|k(v)|(cid:2)c+(6+ε) log|k(v)|, wherevrunsthroughthenonarchimedeanvaluations10ofK correspondingtosingular fibres, and k(v) denote(cid:4)s(cid:2)the finite resi(cid:5)due field of K at v.11 For the curve EK as above,thequantityexp nvlog|k(v)| coincideswitht(cid:2)heabsolutenormN(DiscEK) of the so-called minimal discriminant of E , and exp( log|k(v)|) coincides with K the absolute norm N(Cond ) of the conductor of E .12 Using these notational EK K conventions, the Szpiro conjecture states that if K is a number field, then for every ε >0thereisarealc(cid:7) >0(dependingon K andε)suchthatforeveryellipticcurve E over K theinequality K N(Disc )(cid:2)c(cid:7)N(Cond )6+ε EK EK holds,seee.g.[51,ChapterIV,10.6].13 10 Byabuseofsomeoftheestablishedterminology,valuationsinthistextincludenonarchimedeanand archimedeanones. 11 Thenotation|J|standsforthecardinalityofthesetJ. 12 There are two different objects in this text (and in this subsection) whose names involve the word “discriminant”:theminimaldiscriminantDiscEK ofanellipticcurveEK overanumberfieldK andthe (absolute)discriminantDK/QofanumberfieldK. 13 SzpiroprovedthatoverQthisinequalityimpliestheabcconjectureoverQwithconstant6/5instead ofconstant1init,seee.g.[15,p.598]. 123 NotesonthetheoryofShinichiMochizuki 411 TheVojtaconjecture,asdiscussedinthistext,dealswithasmoothpropergeomet- rically connected curve C over a number field K and a reduced effective divisor D onC suchthatthelinebundleω (D),associatedwiththesumofthedivisor Danda C canonicaldivisorofC,isofpositivedegree(i.e.C\Disahyperboliccurve,see1.5). Itassertsthefollowing:foreverypositiveintegern andpositiverealnumberε there isaconstantc(dependingonC,D,n,εbutnoton K)suchthattheinequality (cid:4) (cid:5) htωC(D)(x)(cid:2)c+(1+ε) log-diffC(x)+log-condD(x) holdsforallx ∈(C \ D)(K(cid:7)),forallnumberfields K(cid:7) ofdegree(cid:2)n.Todefinethe terms,letCbearegularpropermodelofC overSpecO .Forapoint x ∈ C(Qalg) K denotebyFtheminimalsubfieldofQalgoverwhichxisdefined.Lets : SpecO → x F Cbethesectionuniquelydeterminedby x.Thentheheightof x associatedtoaline bundle B onC canbeexplicitlydefinedinseveralequivalentways(uptoabounded functiononC(Qalg)),forinstance,byusingthecanonicalheightonsomeprojective spaceintowhichC isembedded, orasdegs∗B,whereBisanextension toCof B x viewedasanarithmeticlinebundleonC,cf.[31,Section1]or[15,PartB].Define log-cond (x)=deg(s∗D) ,whereDdenotestheclosureinCofD,andredstands D x red forthereducedpart.Definelog-diffC(x) = degδF/Q = |F:Q|−1degDF/Q,where δF/QandDF/QarethedifferentanddiscriminantofF/Q,andthenormaliseddegree degisdefinedin2.2.14Thisconjectureisequivalentto[54,Conjecture2.3]forcurves or[5,Conjecture14.4.10]. UsingtheBelyimap,onereducestheVojtaconjectureforC,D,K asabovetothe caseofC =P1overQand D =[0]+[1]+[∞].15 Note the difference between the Vojta conjecture and the Szpiro conjecture in relation to allowing the algebraic number field to vary; this partially explains the occurrenceoftheterminvolvinglog-diff ontheRHSoftheinequalityoftheformer C conjecture.16 There are also so-called explicit stronger versions of the abc conjecture, which easily imply the Wiles–Fermat theorem, see e.g. [56], and which are not dealt with in[35–38].Foradiscussionoftherelationshipbetween[35–38]andsolutionstothe Fermatequationseethefinalparagraphof2.12. 14 ForafieldextensionR/Sthenotation|R:S|standsforitsdegree. 15 ViewingP1astheλ-lineintheLegendrerepresentationy2=x(x−1)(x−λ)ofanellipticcurveEλ yieldsaclassifyingmorphismfromP1\{0,1,∞}tothenaturalcompactificationMell⊗Qofthemoduli Footnote15continued stackofellipticcurvesoverZtensoredwithQ.TheheighthtωC(D)(λ)ontheLHSoftheVojtaconjecture forC = P1 and D = [0]+[1]+[∞],iscloselyrelatedto1/6timestheLHSoftheinequalityofthe Szpiroconjecturefor Eλ,sincethedegreeofthepull-backtoP1 ofthedivisoratinfinityofthenatural compactificationofMell⊗Qissixtimes1=thedegreeofωC(D),see[31]. 16 OnecanformulateastrongerversionoftheSzpiroconjectureinwhichK varies:foreveryε>0there isaconstantc(cid:7)suchthatthefollowinginequalityholds:N(DiscEK)(cid:2)c(cid:7)N(CondEK)6+ε|DK/Q|6+εfor allellipticcurves EK overnumberfields K.ThisstrongerversionisequivalenttotheVojtaconjecture, asweshallseewhenwemeetitin2.12,anditshowsupinAbstract,thefinalsentenceofSection1and Corollary4.2of[40],andon[39,p.17]. 123 412 IvanFesenko 1.4 Aquestionposedtoastudentbyhisthesisadvisor InJanuary1991ShinichiMochizuki,atthattimeathirdyearPhDstudentinPrinceton, 21 years old, was asked by Faltings (his thesis advisor) to try to prove the effective formoftheMordellconjecture.17 Not surprisingly, he was not able to prove it during his PhD years. As we know, hetooktherequestofhissupervisorveryseriously.Inhindsight,itisastoundingthat almost all his papers are related to the ultimate goal of establishing the conjectures of1.3.Theseeffortsoverthelongtermculminatedtwentyyearslaterin[38],where (a),(c),(d),(e)andhence(b)and(f)of1.3areestablishedasoneapplicationofhis inter-universalTeichmüllertheory[35,37].18 HisearlierHodge–Arakelovtheory[23,24],whereacertainweakarithmeticversion oftheKodaira–Spencermapisstudied,wasalreadyaninnovativestepforward.That workshowsthatGaloisgroupsmayinsomesenseberegardedasarithmetictangent bundles. 1.5 Onanabeliangeometry Algebraic(orétale)fundamentalgroupsingeneralandanabeliangeometryinpartic- ulararelessfamiliartonumbertheoriststhanclassfieldtheoryorpartsofdiophantine geometry.Ontheotherhand,geometersmayfeelmoreathomeinthiscontext.For an introduction to many relevant issues starting with algebraic fundamental groups andleadingtodiscussionsofseveralkeyresultsinanabeliangeometrysee[52,Chap- ter 4]. See also [49] for a survey of several directions in anabelian geometry before 2010,includingdiscussionsofsomeresultsbyMochizukiwhichareprerequisitesfor arithmeticdeformationtheory. Thefactthattheauthorofthistextisnotdirectlyworkinginanabeliangeometrycan beencouragingformanyreadersofthistextwhowouldtypicallysharethisquality. Foranygeometricallyintegral(quasi-compact)scheme X overaperfectfield K, thefollowingexactsequenceisfundamental: 1 → πgeom(X) → π (X) → π (SpecK)=G → 1. 1 1 1 K Hereπ (X)isthealgebraicfundamentalgroupof X,πgeom(X) = π (X× Kalg), 1 1 1 K Kalgisanalgebraicclosureof K,seee.g.[52,Proposition5.6.1].Suppresseddepen- denceofthefundamentalgroupsonbasepointsactuallymeansthatobjectsareoften well-defined only up to conjugation by elements of π (X). Algebraic fundamental 1 groupsofschemesovernumberfields(orfieldscloselyrelatedtonumberfields,such aslocalfieldsorfinitefields)arealsocalledarithmeticfundamentalgroups. 17 Neithertheword“anabelian”northeGrothendieckletter[14]wasmentioned.TheauthorofIUTheard aboutanabeliangeometryforthefirsttimefromTakayukiOdainKyotointhesummerof1992. 18 Thefourparts[35–38]werereadybyAugust2011andputonholdforoneyear.Theywerepostedon theauthor’swebpageinAugust2012andsubmittedtoamathematicaljournal. 123 NotesonthetheoryofShinichiMochizuki 413 IfC isacomplexirreduciblesmoothprojectivecurveminusafinitecollectionof its points, then π (C) is isomorphic to the profinite completion of the topological 1 fundamentalgroupoftheRiemannsurfaceassociatedtoC. IfC istheresultofbase-changingacurveoverafield K tothefieldofcomplex numbers,thentheanalogueforsuchacurveover K ofthedisplayedsequence(asso- ciatedto X)discussedinthepreviousparagraphinducesahomomorphismfromG K tothequotientgroupOut(πgeom(C))oftheautomorphismgroupofπgeom(C)byits 1 1 normalsubgroupofinnerautomorphisms.Belyiproved,usingthetheoremdiscussed in 1.1 for elliptic curves, that this map gives an embedding of the absolute Galois group GQ ofQintotheOutgroupofthepro-finitecompletionofafreegroupwith twogenerators[2].ForreaderswithbackgroundoutsidenumbertheoryIrecallthat, unlike the case with absolute Galois groups of local fields, we still know relatively littleaboutGQ;hencetheBelyiresultisofgreatvalue. RecallthatahyperboliccurveC overafield K ofcharacteristiczeroisasmooth projectivegeometricallyconnectedcurveofgenusgminusrpointssuchthattheEuler characteristic2−2g−r isnegative.Examplesincludeaprojectivelineminusthree points or an elliptic curve minus one point. The algebraic fundamental group of a hyperboliccurveisnonabelian. Anabelian geometry “yoga”for so-called anabelian schemes of finite type over a groundfieldK (suchasanumberfield,afieldfinitelygeneratedoveritsprimesubfield, etc.)statesthatananabelianscheme X canberecoveredfromthetopologicalgroup π (X)andthesurjectivehomomorphismoftopologicalgroupsπ (X)→ G (upto 1 1 K purely inseparable covers and Frobenius twists in positive characteristic). Thus, the algebraicfundamentalgroupsofanabelianschemesarerigid.19 In[14],Grothendieckproposedthefollowingquestions: (a) Arehyperboliccurvesovernumberfieldsorfinitelygeneratedfieldsanabelian? (b) Apointx in X(K),i.e.amorphismSpecK → X,determines,inafunctorial way,acontinuoussectionG →π (X)(well-defineduptocompositionwith K 1 an inner automorphism) of the surjective map π (X) → G . The section 1 K conjecture asks if, for a geometrically connected smooth projective curve X overK,ofgenus>1,themapfromrationalpointsX(K)tothesetofconjugacy classesofsectionsissurjective(injectivitywasalreadyknown).Thereisalso thequestionofwhetherornotthesectionconjecturecouldbeofuseinderiving finitenessresultsindiophantinegeometry. TheNeukirch–Ikeda–Uchidatheoremisabirationalversionof(a)inthelowestdimen- sion.AsimilarrecoverypropertyforfieldsfinitelygeneratedoverQwasprovedby Pop. Later Mochizuki proved a similar recovery property for a subfield of a field 19 Comparewiththefollowingstrongrigiditytheorem(Mostow–Prasad–Gromovrigiditytheorem)for hyperbolic manifolds: the isometry class of a finite-volume hyperbolic manifold of dimension (cid:3) 3 is determinedbyitstopologicalfundamentalgroup,seee.g.[12].Recallthatinétaletopologyopensubschemes ofspectraofringsofintegersofnumberfieldsare,upto2-torsion,of(l-adic)cohomologicaldimension3, seee.g.[22,ChapterII,Theorem3.1]. 123 414 IvanFesenko finitelygeneratedoverQ .Manymoreresultsareknownoverothertypesofground p fields,forasurveyseee.g.[49]. With respect to (a), important contributions were made by Nakamura and Tama- gawa. Then Mochizuki proved that hyperbolic curves over finitely generated fields ofcharacteristiczeroareindeedanabelian.Moreover,usingnonarchimedeanHodge– Tate theory (also called p-adic Hodge theory), Mochizuki proved that a hyperbolic curve X over a subfield K of a field finitely generated over Q can be recovered p functoriallyfromthecanonicalprojectionπ (X)→G . 1 K The section conjecture in part (b) has not been established. A geometric pro- p-version of the section conjecture fails, see [16] and its introduction for more results. A combinatorial version of the section conjecture is established in [18]. It isuncleartowhatextentthesectionconjecturemaybeusefulindiophantinegeom- etry,but[19]proposesamethodwhichmayleadtosuchapplicationsofthesection conjecture. Arithmeticdeformationtheory,thoughrelatedtotheresultsinanabeliangeometry reviewedabove,usesandappliesadifferentsetofconcepts:mono-anabeliangeometry, thenonarchimedeantheta-function,categoriesrelatedtomonoid-theoreticstructures, deconstructionandreconstructionofringstructures. 2 Onarithmeticdeformationtheory The task of presenting arithmetic deformation theory on several pages or in several hoursisaninterestingchallenge.20 InthesenotesIattempttosimplifyasmuchasissensibleandtouseaslittlenew terminologyasisfeasible(andtoindicaterelationswiththeoriginalterminologyof IUTwhenIusedifferentterminology).Asexplainedintheforeword,Iwillhaveto bevaguewhentalkingaboutsomeofthecentralconceptsandobjectsofthetheory. Somemoretechnicalsentenceshavebeenmovedtothefootnotes. 2.1 TextsrelatedtoIUT Inter-universalTeichmüllertheory21hasmanyprerequisitesandoffersmanyinnova- tions. (cid:9) Absolute mono-anabelian geometry, developed in [32–34], is an entralling new theoryinitsownright.Itenhancesanabeliangeometryandbringsittoanewlevel. ItplaysapivotalroleinIUT. (cid:9) The theory of the nonarchimedean theta-function, cf. [30] and a review in [36, Section1],isofsimilarcentralimportanceinIUT. 20 Inviewoftheoverwhelmingnoveltyofthetheory,itishardlypossibletogiveanefficaciouspresentation duringastandardtalk. 21 Thereasonforthisnameiswellexplainedin[35,Introduction],aswellasinthereviewpapers[39,40]. 123