Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta

36 Pages·2015·0.75 MB·English

by  Ivan Fesenko

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