Arithmetic deformation theory via arithmetic fundamental groups and

29 Pages·2017·0.29 MB·English

ARITHMETIC DEFORMATION THEORY VIA
ARITHMETIC FUNDAMENTAL GROUPS AND NONARCHIMEDEAN THETA-FUNCTIONS,
NOTES ON THE WORK OF SHINICHI MOCHIZUKI
IVANFESENKO
ABSTRACT. Thesenotessurveythemainideas,conceptsandobjectsoftheworkbyShinichiMochizukioninter-
universalTeichmüllertheory[31],whichmightalsobecalledarithmeticdeformationtheory,anditsapplication
to diophantine geometry. They provide an external perspective which complements the review texts [32] and
[33]. Some importantdevelopmentswhichpreceded[31]are presentedinthefirstsection. Severalkeyaspects
ofarithmeticdeformationtheoryarediscussedinthesecondsection. Itsmaintheoremgivesaninequality–bound
onthesizeofvolumedeformationassociatedtoacertainlog-theta-lattice. Theapplicationtoseveralfundamental
conjecturesinnumbertheoryfollowsfromafurtherdirectcomputationoftherighthandsideoftheinequality.The
thirdsectionconsidersadditionalrelatedtopics,includingpracticalhintsonhowtostudythetheory.
ThistextwaspublishedinEurop.J.Math.(2015)1:405–440.
CONTENTS
Foreword 2
1. Theorigins 2
1.1. FromclassfieldtheorytoreconstructingnumberfieldstocoveringsofP1 minusthreepoints 2
1.2. Adevelopmentindiophantinegeometry 3
1.3. Conjecturalinequalitiesforthesameproperty 4
1.4. Aquestionposedtoastudentbyhisthesisadvisor 6
1.5. Onanabeliangeometry 6
2. Onarithmeticdeformationtheory 8
2.1. TextsrelatedtoIUT 8
2.2. Initialdata 9
2.3. Abriefoutlineoftheproofandalistofsomeofthemainconcepts 10
2.4. Mono-anabeliangeometryandmultiradiality 11
2.5. Nonarchimedeantheta-functions 13
2.6. GeneralisedKummertheory 14
2.7. Thetheta-linkandtwotypesofsymmetry 15
2.8. Nonarchimedeanlogarithm,log-link,log-theta-lattice,log-shell 17
2.9. Rigiditiesandindeterminacies 18
2.10. Theroleofglobaldata 18
2.11. ThemaintheoremofIUT 20
2.12. TheapplicationofIUT 20
2.13. Moretheorems,objectsandconceptsofIUT 22
2.14. AnalogiesandrelationsbetweenIUTandothertheories 22
3. StudyingIUTandrelatedaspects 24
3.1. OntheverificationofIUT(updatedinJuly2017) 24
1
2 IVANFESENKO
3.2. EntrancestothestudyofIUT(updatedinJuly2017) 25
3.3. TheworkofShinichiMochizuki 25
3.4. Relatedissues 26
References 26
Foreword. Theaimofthesenotesistopresent,inarelativelysimpleandgeneralform,thekeyideas,concepts
andobjectsoftheworkofShinichiMochizukioninter-universalTeichmüllertheory(IUT),toasmanypotential
readersaspossible. ThepresentationisbasedonmyongoingexperienceinstudyingIUT.Thistextisexpected
to help its readers to gain a general overview of the theory and a certain orientation in it, as well as to see
variouslinksbetweenitandexistingtheories.
ReadingthesenotescannotreplaceorsubstituteaseriousstudyofIUT.Thereareprobablytwomainoptions
availableatthetimeofwritingofthistexttolearnabouttheessenceofIUT.Thefirstistodedicateasignificant
amount of time to studying the theory patiently and gradually reaching its main parts. I refer to section 3 for
myrecommendationonhowtostudytheoriginaltextsofarithmeticdeformationtheory,whichisanothername
(duetotheauthorofthepresenttext)forthetheory. Thesecondoptionistoreadthereviewtexts[32]and[33]
andintroductionsofpapers,etc. Myexperienceandtheexperienceofseveralothermathematiciansshowthat
the early review texts that were written by the author of the theory prior to 2015 could be hard to follow, and
readingthembeforeaseriousstudyofIUTmaybenotthebestway,whilereadingthemaftersomepreliminary
studyorinthemiddleofitcanbemoreuseful.
In view of the declared aim of this text and the natural limitation on its size, it is inevitable that several
importantmathematicalobjectsandnotionsinsection2areintroducedinavagueform. Oneofsuchobjectsis
aso-calledtheta-link: averylargepartof[31]-I-IIIdefinesthetheta-linkanddevelopsitsenhancedversions.1
1. THE ORIGINS
1.1. From class field theory to reconstructing number fields to coverings of P1 minus three points.
Abelianclassfieldtheoryforone-dimensionalglobalandlocalfields,inparticularfornumberfieldsandtheir
completions, has played a central role in number theory and stimulated many further developments. Inverse
Galois theory, several versions of the Langlands programme, anabelian geometry, (abelian) higher class field
theory and higher adelic geometry and analysis, and, to some extent, the arithmetic of abelian varieties over
globalfieldsandtheircompletionsareamongthem.
InverseGaloistheorystudieshowtorealisefiniteorinfinitecompacttopologicalgroupsasGaloisgroupsof
various fields including extensions of number fields and 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
everysolublegroupcanberealisedasaGaloisgroupoveraglobalfield,seee.g. §6Ch.IXof[40].
Let Kalg be an algebraic closure of a number field K. The Galois group G = G(Kalg/K) is called the
K
absoluteGaloisgroupofK.
TheNeukirch–Ikeda–Uchidatheorem(provedbytheendof1970s;theproofusedglobalclassfieldtheory)
asserts,seee.g. §2Ch.XIIof[40],thefollowing:
1Anappreciationofthegeneralqualitativeaspectsofthetheta-linkmaybeobtainedbystudyingthesimplestversionofthetheta-
link. Thisversionisdiscussedin§I1of[31]-I,whiletechnicaldetailsconcerningtheconstructionofthisversionmaybefoundin
approximately30pagesof§3of[31]-I.
NOTESONTHEWORKOFSHINICHIMOCHIZUKIONARITHMETICDEFORMATIONTHEORY 3
∼
FortwonumberfieldsK ,K andanyisomorphismoftopologicalgroupsψ: G →− G there
1 2 K1 K2
isauniquefieldisomorphismσ: Kalg→−∼ Kalgsuchthatσ(K )=K andψ(g)(a)=σ−1(gσ(a))
2 1 2 1
foralla∈K2alg,g∈GK1. Inparticular,thehomomorphismGQ→Aut(GQ),g(cid:55)→(h(cid:55)→ghg−1),
isanisomorphismandeveryautomorphismofGQ isinner.
ThenexttheoremwasincludedasTh. 4in[3]in1980,aspartofBelyi’sstudyofaspectsofinverseGalois
theory:
An irreducible smooth projective algebraic curveC defined over a field of characteristic zero
can be defined over an algebraic closure Qalg of the field of rational numbers Q if and only if
thereisacoveringC−→P1 whichramifiesovernomorethanthreepointsofP1.
Thistheorem2playsakeyroleinthestudyofGaloisgroups,includingalgebraicandgeometricfundamental
groups of curves over number fields and local fields.3 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,afundamen-
talfinitenesspropertyindiophantinegeometry[8],[2]. TheFaltings–Mordelltheoremassertsthat
AcurveCofgenus>1definedoveranalgebraicnumberfieldKhasonlyfinitelymanyrational
pointsoverK.
Several other proofs followed.4 Vojta found interesting links with Nevanlinna theory in complex analysis
whichledtooneofhisproofs. Fortextbookexpositionsofsimplifiedproofssee[15]and[6].
ThesameyearGrothendieckwrotealetter[13]toFaltings,whichproposedelementsofanabeliangeometry.
With hindsight, one of the issues raised in it was a generalisation of the Neukirch–Ikeda–Uchida theorem5 to
anabeliangeometricobjectssuchashyperboliccurvesovernumberfieldsandpossibleapplicationsofanabelian
geometry to provide new proofs and stronger versions, as well as a better understanding, of such results in
diophantinegeometryastheFaltings–Mordelltheorem,cf. 1.5.
2 The first version of [3] and Belyi’s seminar talk in Moscow dealt with an elliptic curve, which was enough for its subsequent
application, see1.5. Afterreadingthefirstversionof[3], Bogomolovnoticedthattheoriginalproofofthetheoreminitworksfor
arbitrarycurves.HetoldBelyihowimportantthisextendedversionisandurgedhimtoincludetheextendedversioninthepaper.Belyi
wasreluctanttoincludetheextendedversion,onthegroundsthatoverfinitefieldseveryirreduciblesmoothprojectivealgebraiccurve
maybeexhibitedasacoveringoftheprojectivelinewithatmostoneramificationpoint.BogomolovthentalkedwithShafarevich,who
immediatelyappreciatedthevalueoftheextendedversionandinsistedthattheauthorincludeitin[3]. Bogomolovfurtherdeveloped
thetheoryofBelyimaps,inparticular,inrelationtotheuseofcolouredRiemannsurfacesanddeliverednumeroustalksonthesefurther
developments.Severalyearslaterthistheoryappeared,independently,inGrothendieck’stext[12].
3Grothendieckwroteaboutthe“onlyif”part: “never,withoutadoubt,wassuchadeepanddisconcertingresultprovedinsofew
lines!”[12].
4Hereweareinthebestpossiblesituationwhenaconjecturestatedoveranarbitraryalgebraicnumberfieldandisestablishedover
anarbitraryalgebraicnumberfield, andthemethodsoftheproofsdonotdependonthespecificfeaturesofthenumberfieldunder
consideration.ThisisnotsointhecaseofthearithmeticLanglandscorrespondence,evenforellipticcurvesovernumberfields.Inthe
historyofclassfieldtheory,theinitialperiodofdevelopingspecialtheoriesthatworkonlyoversmallnumberfieldswasfollowedby
aphaseofgeneralfunctorialclassfieldtheoryoverarbitraryglobalandlocalfields. Thegeneraltheorywaseventuallyclarifiedand
simplified,see[39],anditbecameeasierthanthoseinitialtheories. Weareyettowitnessasimilarphasewhichinvolvesageneral
functorialtheorythatworksoverarbitrarynumberfieldsinthecaseoftheLanglandsprogrammeandhence,inparticular,yieldsanother
proofoftheWiles–Fermattheoremviatheautomorphicpropertiesofellipticcurvesoveranynumberfield.
5itappearsthatGrothendieckwasnotawareofthistheorem
4 IVANFESENKO
1.3. Conjecturalinequalitiesforthesameproperty. Thereareseveralcloselyrelatedconjectures,proposed
intheperiodfrom1978to1987,whichextendfurtherthepropertystatedintheMordellconjecture:
(a)theeffectiveMordellconjecture—aconjecturalextensionoftheFaltings–Mordelltheoremwhichinvolves
aneffectiveboundontheheightofrationalpointsofthecurveCoverthenumberfieldKintheFaltingstheorem
intermsofdataassociatedtoCandK,
(b)theSzpiroconjecture,seebelow,
(c) the Masser–Oesterlé conjecture, a.k.a. the abc conjecture (whose statement over Q is well known6, and
whichhasanextensiontoarbitraryalgebraicnumberfields,seeConj. 14.4.12of[6]),
(d)theFreyconjecture,seeConj. F.3.2(b)of[15],
(e)theVojtaconjectureonhyperboliccurves,seebelow,
(f)arithmeticBogomolov–Miyaoka–Yauconjectures(thereareseveralversions).
The Szpiro conjecture was stated several years before7 the work of Faltings, who learned much about the
subject related to his proof from Szpiro. Using the Frey curve8, it is not difficult to show that (c) and (d) are
equivalentandthattheyimply(b),seee.g. seesect. F3of[15]andreferencestherein. UsingBelyimapsasin
1.1,onecanshowtheequivalenceof(c)and(a). Fortheequivalenceof(c)and(e)seee.g. Th. 14.4.16of[6]
and[47]. Forimplications(e)⇒(f)see[48].
OverthecomplexnumbersthepropertyanalogoustotheSzpiroconjectureisveryinteresting. Forasmooth
projectivesurfaceequippedwithastructureofnon-splitminimalellipticsurfacefibredoverasmoothprojective
connected complex curve of genus g, such that the fibration admits a global section, and, moreover, every
singularfibreofthefibrationisoftypeI ,i.e. itscomponentsareprojectivelineswhichintersecttransversally
n
and form an n-gon, this property states that the sum of the number of components of singular fibres does
not exceed 6 times the sum of the number of singular fibres and of 2g−2. The property has several proofs,
of which the first was given by Szpiro. Shioda deduced the statement in two pages of computations from
arguments already known to Kodaira. These two proofs of the geometric version of the inequality use the
cotangentbundleandtheKodaira–Spencermap. A(full)arithmeticversionoftheKodaira–Spencermapcould
be quite useful for giving a proof of the arithmetic Szpiro conjecture. However, such an arithmetic version of
theKodaira–Spencermapisnotyetknown.
Among several other proofs of this property, a proof by Bogomolov uses monodromy actions and the hy-
perbolic geometry of the upper half-plane and does not use the cotangent bundle, see sect. 5.3 of [1]. His
proofmakesessentialuseofthefactthatthen-gonsdeterminedbythesingularfibresmaybeequippedwitha
common orientation, like windmills revolving in synchrony in the presence of wind. Synchronisation of data
plays an important role in arithmetic deformation theory as well, cf. 2.10. To develop an arithmetic analogue
ofthegeometricproofofBogomolovtoapplytoprovingthearithmeticSzpiroconjecture,oneneedsakindof
arithmeticanalogueofthehyperbolicgeometryoftheupperhalf-plane, andthisisinsomesenseachievedby
IUT,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
K
6Foreveryε>0thereisaconstantsuchthatforallnon-zerointegersa,b,csuchthat(a,b,c)=1theequalitya+b+c=0implies
max(log|a|,log|b|,log|c|)(cid:54)constant+(1+ε)∑p|abclogpwhereprunsthroughallpositiveprimesdividingabc.Whilethestatement
oftheabcconjecturedoesnotrevealimmediatelyanyunderlyinggeometricstructure,theotherconjecturesaremoregeometrical.For
anentertainingpresentationofaspectsoftheabcconjectureandrelatedproperties,seee.g.[49].
7In1978Szpirotalkedaboutitwithseveralmathematicians. HemadetheconjecturepublicatameetingoftheGermanMath.
Society(DMV)in1982whereFrey,OesterléandMasserwerepresent.
8y2=x(x+a)(x−b)wherea,b,a+barenon-zerocoprimeintegers
NOTESONTHEWORKOFSHINICHIMOCHIZUKIONARITHMETICDEFORMATIONTHEORY 5
withsplitmultiplicativereductionateverybadreductionvaluation,soallsingularfibresofitsminimalregular
proper model E −→SpecO are of type I , the weighted sum of the numbers n of components of singular
K n v
fibressatisfies
∑n log|k(v)|(cid:54)c+(6+ε)∑log|k(v)|,
v
where v runs through the nonarchimedean valuations9 of K corresponding to singular fibres, and k(v) denotes
thefiniteresiduefieldofKatv.10 ForthecurveEK asabove,thequantityexp(∑nvlog|k(v)|)coincideswiththe
absolutenormN(DiscEK)oftheso-calledminimaldiscriminantofEK,andexp(∑log|k(v)|)coincideswiththe
absolutenormN(Cond )oftheconductorofE .11 Usingthesenotationalconventions,theSzpiroconjecture
EK K
statesthatifK isanumberfield,thenforeveryε >0thereisarealc(cid:48)>0(dependingonK andε)suchthatfor
everyellipticcurveE overK theinequality
K
N(Disc )(cid:54)c(cid:48)N(Cond )6+ε
EK EK
holds,seee.g. 10.6ofCh.IVof[44].12
TheVojtaconjecture,asdiscussedinthistext,dealswithasmoothpropergeometricallyconnectedcurveC
overanumberfieldK andareducedeffectivedivisorDonC suchthatthelinebundleω (D),associatedwith
C
the sum of the divisor D and a canonical divisor ofC, is of positive degree (i.e. C\D is a hyperbolic curve,
see 1.5). It asserts the following: for every positive integer n and positive real number ε there is a constant c
(dependingonC,D,n,ε butnotonK)suchthattheinequality
ht (x)(cid:54)c+(1+ε)(log-diff (x)+log-cond (x))
ωC(D) C D
holds for all x∈(C\D)(K(cid:48)), for all number fields K(cid:48) of degree (cid:54)n. To define the terms, let C be a regular
propermodelofC overSpec(O ). Forapointx∈C(Qalg)denotebyF aminimalsubfieldofQalg overwhich
K
xisdefined. Lets : Spec(O )−→C bethesectionuniquelydeterminedbyx. Thentheheightofxassociated
x F
to a line bundle B on C can be explicitly defined in several equivalent ways (up to a bounded function on
C(Qalg)), for instance, by using the canonical height on some projective space into whichC is embedded, or
as deg(s∗B), where B is an extension to C of B viewed as an arithmetic line bundle on C, cf. sect. 1 of
x
[29]orpartBof[15]. Definelog-cond (x)=deg((s∗D) ),whereD denotestheclosureinC ofD,andred
D x red
stands for the reduced part. Define the log-different (also called the normalised log-absolute-discriminant) as
log-diffC(x)=deg(δF/Q)=|F :Q|−1deg(DF/Q) where δF/Q and DF/Q are the different and discriminant of
F/Q,andthenormaliseddegreedegisdefinedin2.2.13 ThisconjectureisequivalenttoConj. 2.3forcurvesin
[47]orConj. 14.4.10in[6].
UsingtheBelyimap,onereducestheVojtaconjectureforC,D,K asabovetothecaseofC=P1 overQand
D=[0]+[1]+[∞].14
9byabuseofsomeoftheestablishedterminology,valuationsinthistextincludenonarchimedeanandarchimedeanones
10thenotation|J|standsforthecardinalityofthesetJ
11therearetwodifferentobjectsinthistext(andinthissubsection)whosenamesinvolvetheword“discriminant”: theminimal
discriminantDiscEK ofanellipticcurveEK overanumberfieldKandthe(absolute)discriminantDK/QofanumberfieldK
12SzpiroprovedthatoverQthisinequalityimpliestheabcconjectureoverQwithconstant6/5insteadofconstant1init,seee.g.
p.598of[15].
13forafieldextensionR/Sthenotation|R:S|standsforitsdegree
14ViewingP1astheλ-lineintheLegendrerepresentationy2=x(x−1)(x−λ)ofanellipticcurveE yieldsaclassifyingmorphism
λ
fromP1\{0,1,∞}tothenaturalcompactificationM ⊗QofthemodulistackofellipticcurvesoverZtensoredwithQ. Theheight
ell
ht (λ)ontheLHSoftheVojtaconjectureforC=P1andD=[0]+[1]+[∞],iscloselyrelatedto 1 timestheLHSoftheinequality
ωC(D) 6
of the Szpiro conjecture for E , since the degree of the pull-back to P1 of the divisor at infinity of the natural compactification of
λ
Mell⊗Qis6times1=thedegreeofωC(D),see[29].
6 IVANFESENKO
NotethedifferencebetweentheVojtaconjectureandtheSzpiroconjectureinrelationtoallowingthealge-
braicnumberfieldtovary;thispartiallyexplainstheoccurrenceoftheterminvolvinglog-diff ontheRHSof
C
theinequalityoftheformerconjecture. OnecanformulateastrongerversionoftheSzpiroconjectureinwhich
K varies.15 TheFreyconjecture,theSzpiroconjectureandthestrongerSzpiroconjectureareproblemsinarith-
meticofellipticcurvesovernumberfields. Therearethreebasicmethodstoworkonfundamentalproblemsin
arithmeticofellipticcurves: (i)themoretraditionaluseofGaloisgroupsgeneratedbycertaintorsionpointsof
theellipticcurveE andassociatedGaloisrepresentations,inparticular,thisishowtheWiles–Fermattheorem
K
wasproved,(ii)themethodofIUTwhichinvolvesthearithmeticfundamentalgroupofhyperboliccurvesover
number fields related to the elliptic curve E , among several ingredients, (iii) higher adelic method uses the
K
abelian part of the absolute Galois group of the function field of the elliptic curve E , two-dimensional class
K
fieldtheory,higheradelesandhigherzetaintegrals,[9]and2.14.
Therearealsoso-calledexplicitstrongerversionsoftheabcconjecture,whicheasilyimplytheWiles–Fermat
theorem, seee.g. [49], and which are not dealtwith in[31]. Fora discussionof therelationship between[31]
andsolutionstotheFermatequationseethefinalparagraphof2.12.
1.4. Aquestionposedtoastudentbyhisthesisadvisor. InJanuary1991ShinichiMochizuki,atthattimea
thirdyearPhDstudentinPrinceton,21yearsold,wasaskedbyFaltings(histhesisadvisor)totrytoprovethe
effectiveformoftheMordellconjecture.16
Not surprisingly, he was not able to prove it during his PhD years. As we know, he took the request of his
supervisorveryseriously. Inhindsight,itisastoundingthatalmostallhispapersarerelatedtotheultimategoal
ofestablishingtheconjecturesof1.3. Theseeffortsoverthelongtermculminatedtwentyyearslaterin[31]-IV,
where (a), (c), (d), (e) and hence (b) and (f) of 1.3 are established as one application of his inter-universal
Teichmüllertheory[31]-I-III.17
His earlier Hodge–Arakelov theory [23], where a certain weak arithmetic version of the Kodaira–Spencer
mapisstudied,wasalreadyaninnovativestepforward. ThatworkshowsthatGaloisgroupsmayinsomesense
beregardedasarithmetictangentbundles.
1.5. On anabelian geometry. Algebraic (or étale) fundamental groups in general and anabelian geometry
in particular are less familiar to number theorists than class field theory or parts of diophantine geometry.
On the other hand, geometers may feel more at home in this context. For an introduction to many relevant
issuesstartingwithalgebraicfundamentalgroupsandleadingtodiscussionsofseveralkeyresultsinanabelian
geometryseeCh.4of[45]. Seealso[42]forasurveyofseveraldirectionsinanabeliangeometrybefore2010,
includingdiscussionsofsomeresultsbyMochizukiwhichareprerequisitesforarithmeticdeformationtheory.
The fact that the author of this text is not directly working in anabelian geometry can be encouraging for
manyreadersofthistextwhowouldtypicallysharethisquality.
Foranygeometricallyintegral(quasi-compact)schemeX overaperfectfieldK,thefollowingexactsequence
isfundamental
1→πgeom(X)→π (X)→π (Spec(K))=G →1.
1 1 1 K
15Foreveryε>0thereisaconstantc(cid:48) suchthatthefollowinginequalityholdsN(DiscEK)(cid:54)c(cid:48)N(CondEK)6+ε|DK/Q|6+ε forall
ellipticcurvesEK overnumberfieldsK.ThisstrongerversionisequivalenttotheVojtaconjecture,asweshallseewhenwemeetitin
2.12,anditshowsupinAbstract,thefinalsentenceof§1andCorollary4.2of[33],andonp.17of[32].
16Neithertheword“anabelian”northeGrothendieckletter[13]wasmentioned.TheauthorofIUTheardaboutanabeliangeometry
forthefirsttimefromTakayukiOdainKyotointhesummerof1992.
17Thefourpartsof[31]werereadybyAugust2011andputonholdforoneyear. Theywerepostedontheauthor’swebpagein
August2012andsubmittedtoamathematicaljournal.
NOTESONTHEWORKOFSHINICHIMOCHIZUKIONARITHMETICDEFORMATIONTHEORY 7
Hereπ (X)isthealgebraicfundamentalgroupofX,πgeom(X)=π (X× Kalg),Kalgisanalgebraicclosureof
1 1 1 K
K,seee.g. Prop.5.6.1of[45]. Suppresseddependenceofthefundamentalgroupsonbasepointsactuallymeans
thatobjectsareoftenwell-definedonlyuptoconjugationbyelementsofπ (X). Algebraicfundamentalgroups
1
ofschemesovernumberfields(orfieldscloselyrelatedtonumberfields,suchaslocalfieldsorfinitefields)are
alsocalledarithmeticfundamentalgroups.
IfC is a complex irreducible smooth projective curve minus a finite collection of its points, then π (C) is
1
isomorphictotheprofinitecompletionofthetopologicalfundamentalgroupoftheRiemannsurfaceassociated
toC.
IfC istheresultofbase-changingacurveoverafieldK tothefieldofcomplexnumbers,thentheanalogue
forsuchacurveoverKofthedisplayedsequence(associatedtoX)discussedinthepreviousparagraphinduces
ahomomorphismfromG tothequotientgroupOut(πgeom(C))oftheautomorphismgroupofπgeom(C)byits
K 1 1
normalsubgroupofinnerautomorphisms. Belyiproved,usingthetheoremdiscussedin1.1forellipticcurves,
that this map gives an embedding of the absolute Galois group GQ of Q into the Out group of the pro-finite
completionofafreegroupwithtwogenerators[3]. ForreaderswithbackgroundoutsidenumbertheoryIrecall
that,unlikethecasewithabsoluteGaloisgroupsoflocalfields,westillknowrelativelylittleaboutGQ;hence
theBelyiresultisofgreatvalue.
Recall that a hyperbolic curveC over a field K of characteristic zero is a smooth projective geometrically
connected curve of genus g minus r points such that the Euler characteristic 2−2g−r is negative. Examples
include a projective line minus three points or an elliptic curve minus one point. The algebraic fundamental
groupofahyperboliccurveisnonabelian.
Anabelian geometry “yoga” for so-called anabelian schemes of finite type over a ground field K (such as a
numberfield,afieldfinitelygeneratedoveritsprimesubfield,etc.) statesthatananabelianschemeX canbere-
coveredfromthetopologicalgroupπ (X)andthesurjectivehomomorphismoftopologicalgroupsπ (X)→G
1 1 K
(uptopurelyinseparablecoversandFrobeniustwistsinpositivecharacteristic). Thus,thealgebraicfundamen-
talgroupsofanabelianschemesarerigid.18
In[13],Grothendieckproposedthefollowingquestions:
(a)Arehyperboliccurvesovernumberfieldsorfinitelygeneratedfieldsanabelian?
(b)ApointxinX(K),i.e. amorphismSpec(K)−→X,determines,inafunctorialway,acon-
tinuoussectionG →π (X)(well-defineduptocompositionwithaninnerautomorphism)of
K 1
thesurjectivemapπ (X)→G . Thesectionconjectureasksif,forageometricallyconnected
1 K
smoothprojectivecurveX overK,ofgenus>1,themapfromrationalpointsX(K)totheset
ofconjugacyclassesofsectionsissurjective(injectivitywasalreadyknown). Thereisalsothe
question of whether or not the section conjecture could be of use in deriving finiteness results
indiophantinegeometry.
TheNeukirch–Ikeda–Uchidatheoremisabirationalversionof(a)inthelowestdimension. Asimilarrecov-
erypropertyforfieldsfinitelygeneratedoverQwasprovedbyPop. LaterMochizukiprovedasimilarrecovery
property for a subfield of a field finitely generated over Q . Many more results are known over other types of
p
groundfields,forasurveyseee.g. [42].
18Comparewiththefollowingstrongrigiditytheorem(Mostow–Prasad–Gromovrigiditytheorem)forhyperbolicmanifolds: the
isometryclassofafinite-volumehyperbolicmanifoldofdimension(cid:62)3isdeterminedbyitstopologicalfundamentalgroup,seee.g.
[11]. Recallthatinétaletopologyopensubschemesofspectraofringsofintegersofnumberfieldsare, upto2-torsion, of(l-adic)
cohomologicaldimension3,seee.g.Th.3.1Ch.IIof[22].
8 IVANFESENKO
With respect to (a), important contributions were made by Nakamura and Tamagawa. Then Mochizuki
proved that hyperbolic curves over finitely generated fields of characteristic zero are indeed anabelian. More-
over, using nonarchimedean Hodge–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 functorially from
p
thecanonicalprojectionπ (X)→G .
1 K
The section conjecture in part (b) has not been established. A geometric pro-p-version of the section con-
jecturefails,see[16]anditsintroductionformoreresults. Acombinatorialversionofthesectionconjectureis
established in [18]. It is unclear to what extent the section conjecture may be useful in diophantine geometry,
but[19]proposesamethodwhichmayleadtosuchapplicationsofthesectionconjecture.
Arithmeticdeformationtheory,thoughrelatedtotheresultsinanabeliangeometryreviewedabove,usesand
applies a different set of concepts: mono-anabelian geometry, the nonarchimedean theta-function, categories
relatedtomonoid-theoreticstructures,deconstructionandreconstructionofringstructures.
2. ON ARITHMETIC DEFORMATION THEORY
The task of presenting arithmetic deformation theory on several pages or in several hours is an interesting
challenge.19
InthesenotesIattempttosimplifyasmuchasissensibleandtouseaslittlenewterminologyasisfeasible
(andtoindicaterelationswiththeoriginalterminologyofIUTwhenIusedifferentterminology). Asexplained
in the foreword, I will have to be vague when talking about some of the central concepts and objects of the
theory. Somemoretechnicalsentenceshavebeenmovedtothefootnotes.
2.1. TextsrelatedtoIUT. Inter-universalTeichmüllertheory20hasmanyprerequisitesandoffersmanyinno-
vations.
∗ Absolute mono-anabelian geometry, developed in [30], is an entralling new theory in its own right. It
enhancesanabeliangeometryandbringsittoanewlevel. ItplaysapivotalroleinIUT.
∗ The theory of the nonarchimedean theta-function, cf. [28] and a review in §1 of [31]-II, is of similar
centralimportanceinIUT.
∗ Categorical geometry papers discuss the theory of categories associated to monoid-theoretic struc-
tures21,suchasfrobenioids22[27],aswellasthetheoryofanabelioids[24],[26].23
∗ [31]-I-IIIintroduceandstudyseveralversionsofthetheta-link. Thekeymaintheoremofthefirstthree
partsofIUTisstatedinCor. 3.12of[31]-III.
19Inviewoftheoverwhelmingnoveltyofthetheory,itishardlypossibletogiveanefficaciouspresentationduringastandardtalk.
20ThereasonforthisnameiswellexplainedintheIntroductionof[31]-I,aswellasinthereviewpapers[32]and[33].
21 The term “monoid-theoretic” in this text corresponds to the term “frobenius-like” in [31]. In IUT, the underlying abstract
topologicalgroupsassociatedtoétalefundamentalgroupsareoftenreferredtoasétale-likestructures,see[27]-I,[30]-III.Étale-like
structuresarefunctorial,rigidandinvariantwithrespecttothelinksinIUT,whilefrobenius-likestructuresareusedtoconstructthe
links.Thesituationwhichservesasasortoffundamentalmodelforthetermsfrobenius-likeandétale-likeistheinvarianceoftheétale
sitewithrespecttotheFrobeniusmorphisminpositivecharacteristic,seeExample3.6of[31]-IV.Relationsbetweenthesetwotypesof
structuresarecrucial.Suchrelationsarepresentedfurtherinthesenoteswithoutusingtheterminologyoffrobenius-likeandétale-like.
22ThetheoryoffrobenioidsismotivatedbytheneedtodevelopageometrybuiltupsolelyfromGaloistheoryandmonoid-theoretic
structuresinwhichakindofFrobeniusmorphismonnumberfields,whichdoesnotexistintheusualsense,canbeconstructed. The
availabilityofsuchFrobeniusmorphismsinthetheoryoffrobenioidsleadstovariousanalogiesbetweenIUTand p-adicTeichmüller
theory.Fortwoexamplesoffrobenioidssee2.10.
23 These papers contain much more material than is necessary for the purposes of IUT. If one understands the philosophy that
underliesthesepapers,itispossibletoskiplongtechnicalproofs.
NOTESONTHEWORKOFSHINICHIMOCHIZUKIONARITHMETICDEFORMATIONTHEORY 9
∗ Strengthened versions of notion of a Belyi map obtained in [25] are applied in [29] to prove a new
interestingequivalentformoftheVojtaconjecture,whichisstudiedin[31]-IV.
∗ A straightforward computation of the objects that appear in the main theorem of [31]-I-III is sum-
marisedinTh. 1.10of[31]-IV.InCor. 2.2of[31]-IV,onemakesachoiceofacertainprimenumberl
whichappearsinthiscomputation. ThisleadstotheapplicationtothenewformoftheVojtaconjecture
andhencetotheconjecturesin1.3overanynumberfield.
See3.2forasuggestionofpossibleentriesintothetheory.
2.2. Initial data. There are several equivalent ways to define a normalised degree deg. I will use adeles.
Recall that there is a canonical surjective homomorphism from the group A× of ideles of a number field k to
k
the group Div of complete (i.e. involving archimedean data) divisors associated to k. This group Div may
k k
be described as the direct sum of value groups associated to the nonarchimedean and archimedean valuations
ofk. Thus,suchavaluegroupisisomorphictoZifthevaluationisnonarchimedeanandtoRifthevaluation
is archimedean. Similarly, there are canonical surjective homomorphisms from A× to the group of complete
k
divisorclassesassociatedtok,tothegroupofisomorphismclassesofcompletelinebundlesonSpec(O )and
k
to the group I of fractional ideals of k. For a number field k and an idele α ∈A× define its (non-normalised)
k k
degree deg as −log|α|, where |α| is the canonical module associated to the adelic ring as a locally compact
k
ring by the standard formula |α|=µ(αA)/µ(A), and A is any measurable subset of A of non-zero measure
k
with respect to any nontrivial translation invariant measure µ on the underlying additive group of A . Then
k
degk(α)=degQ(Nk/Qα),andthedegreeofthediagonalimageofanelementofk× inA×k is0.
Duetotheminussigninthedefinitionofdeg , itisminusthenon-normaliseddegreewhichcanbeviewed
k
as the log-volume of αA, where A is, say, the product of the closed balls of radius 1 with centre at 0 for all
completionsofk,and µ isnormalisedtogiveAlog-volume0.
Write limA× for the inductive limit, with respect to the inclusions induced by field embeddings, of the
−→ k
groups of ideles of all finite extensions k of Q in a fixed algebraic closure Qalg. For β ∈limA×, define its
−→ k
normaliseddegreedeg(β)as|k:Q|−1deg (β),wherekisanyalgebraicnumberfieldsuchthatβ corresponds
k
toanelementofA×. Oneverifiesimmediatelythatthisdefinitiondoesnotdependonk. Finally,foranelement
k
γ of the perfection of limA× define its normalised degree deg(γ) as n−1deg(γn), where n(cid:62)1 is any integer
−→ k
suchthatγn ∈limA×. GivenafractionalidealinI , acompletedivisorinDiv , acompletedivisorclass, ora
−→ k k k
linebundle,thenormaliseddegreeofanyofitsliftstothegroupofidelesdoesnotdependonthechoiceoflift
(since the local components of such lifts are completely determined up to unit multiples). Denote this degree
bythesamenotationdeg.
LetE beanellipticcurveoveranumberfieldF withsplitmultiplicativereduction. Ifvisabadreduction
F
valuation and F is the completion of F with respect to v, then the Tate curve F×/(cid:104)q (cid:105), where q is the q-
v v v v
parameterofE atvand(cid:104)q (cid:105)isthecyclicgroupgeneratedbyq ,isisomorphictoE (F),(cid:104)q (cid:105)(cid:55)→theoriginof
F v v F v v
E ,seeCh.Vof[44]and§5Ch.IIof[43].
F
Assume further that the 6-torsion points of E are rational over F, and F contains a 4th primitive root i of
F
unity.
OneworkswiththehyperboliccurveX =E \{0}overF andthehyperbolicorbicurveC =X /±1over
F F F F
F obtainedbyformingthestack-theoreticquotientofX bytheuniqueF-involution−1ofX .
F F
IfkisafieldextensionofF,thendenoteE =E × k,X =X × k,C =C × k.24
k F F k F F k F F
24ForbadreductionvaluationsonealsoworkswithaninfiniteZ-(tempered)coveringYvofamodelXFv ofXFv whichcorresponds
tothekernelofthenaturalsurjectionfromthetemperedfundamentalgrouptoZassociatedtotheuniversalgraph-coveringofthedual
graphofthespecialfibreofXFv.ThespecialfibreofYvisaninfinitechainofcopiesofP1joinedat0and∞.
10 IVANFESENKO
Define an idele q ∈A : its components at archimedean and good reduction valuations are taken to be
EF F
1; its components at bad valuations are taken to be q , where q is the q-parameter of the Tate elliptic curve
v v
E (F)=F×/(cid:104)q (cid:105). Thenumbern ofcomponentsofE atabadreductionvaluationvisexactlythevalueof
F v v v v F
the surjective discrete valuation v: F× →Z at q . Thus, deg (q ) is the LHS of the inequality of the Szpiro
v v F EF
conjecturein1.3. Theultimategoalofthetheoryistogiveasuitableboundfromaboveondeg(q ).
EF
2.3. A brief outline of the proof and a list of some of the main concepts. Conventional scheme-theoretic
geometry is insufficient for the purposes of arithmetic deformation theory. This is one of the reasons why it
was not developed earlier. IUT goes beyond standard arithmetic geometry. Still, it remains quite geometric
and categorical. In its application to the conjectures of 1.3 it does not need to use more from analytic number
theorythantheprimenumbertheorem.
Fix a prime integer l >3 which is relatively prime to the bad reduction valuations of E , as well as to the
F
valuen ofthelocalsurjectivediscretevaluationoftheq-parameterq foreachbadreductionvaluationv.
v v
In 2.12, l will be chosen to be relatively large, so in IUT one often views Z/lZ as a kind of approximation
toZ,see§1.3of[23]formoreonthis.25
AssumethattheextensionKofF generatedbythel-torsionpointsofE hasGaloisgroupoverF isomorphic
F
toasubgroupofGL (Z/lZ)whichcontainsSL (Z/lZ).26
2 2
Due to various reasons motivated by Hodge–Arakelov theory, cf. §1 of [32], §1 of [33], it makes a lot of
sense to look at the monoid-theoretic maps defined, for bad reduction valuations v, on the submonoid of the
multiplicativegroupF× generatedbyunitsandq asfollows:
v v
q (cid:55)−→qm2, u(cid:55)−→u forallu∈O×,
v v Fv
whereO istheringofintegersofF,misafixedintegersuchthat1(cid:54)m(cid:54)(l−1)/2.
Fv v
Theelementqm2 willbeviewedasaspecialvalueofacertainnonarchimedeantheta-function.
v
Choose a 2lth root q of q. We are now led to the study of a monoid-theoretic map which forms part of a
so-calledtheta-link,andwhichatbadreductionvaluationscanbeviewedastheassignment
q (cid:55)−→ (cid:8)Θ(cid:0)(cid:112)−qm(cid:1)=qm2(cid:9) .
1(cid:54)m(cid:54)(l−1)/2
Thismapisnotscheme-theoretic. Itsapplicationmaybeviewedasadeconstructionoftheringstructure.27 To
reconstruct the ring structure, one uses generalised Kummer theory (cf. 2.6), two types of symmetry (cf. 2.7),
rigidities(cf. 2.9)andsplittings(cf. 2.7),allofwhicharecloselyrelatedtothetheta-link(cf. 2.7).
In order to reconstruct portions of the ring structure related to the theta-link, it is necessary to make use of
(archimedean and nonarchimedean) logarithms, in the form of a so-called log-link, cf. 2.8. The theory of the
log-link also involves the mono-anabelian geometry, cf. 2.4, developed in [30]-III. Moreover, one must make
useofinfinitelymanylog-links.
Various copies of the theta-link will form horizontal arrows between two vertical lines formed by the log-
linksofalog-theta-lattice.
25WhentheprimenumberlischoseninCor. 2.2of[31]-IV,someoftheseconditionsonlmaybeslightlyweakened,bytreating
certainbadreductionvaluationsofEF asiftheyaregoodreductionvaluations.
26OnealsoassumesthatCK isaterminalobjectinthecategorywhoseobjectsaregenericallyscheme-likealgebraicstacksZthat
admitafiniteétalemorphismtoCK,andwhosemorphismsarefiniteétalemorphismsofstacksZ1−→Z2definedoverK(thatdonot
necessarilylieoverCK),thisassumptionimpliesthatCF hasauniquemodeloverthefieldFmoddefinedin2.6,c.f.Rk3.1.7(i)of[31]-I.
27Thismapwillbediscussedfurtherin2.7.