Introduction to modern number theory = 现代数论导引 / 2nd ed.

partiproblemsandtricks.

elementarynumbertheory

1.1problemsaboutprimes.divisibilityandprimality

1.1.1arithmeticalnotation

1.1.2primesandcompositenumbers

1.1.3thefactorizationtheoremandtheeuclidean

algorithm

1.1.4calculationswithresidueclasses

1.1.5thequadraticreciprocitylawanditsuse

1.1.6thedistributionofprimes

1.2diophantineequationsofdegreeoneandtwo

1.2.1theequationax+by=c

1.2.2lineardiophantinesystems

1.2.3equationsofdegreetwo

1.2.4theminkowski-hasseprincipleforquadraticforms

1.2.5pell'sequation

1.2.6representationofintegersandquadraticformsby

quadraticforms

1.2.7analyticmethods

1.2.8equivalenceofbinaryquadraticforms

.1.3cubicdiophantineequations

1.3.1theproblemoftheexistenceofasolution

1.3.2additionofpointsonacubiccurve

1.3.3thestructureofthegroupofrationalpointsofa

non-singularcubiccurve

1.3.4cubiccongruencesmoduloaprime

1.4approximationsandcontinuedfractions

1.4.1bestapproximationstoirrationalnumbers

1.4.2fareyseries

1.4.3continuedfractions

1.4.4sl2-equivalence

1.4.5periodiccontinuedfractionsandpell'sequation

1.5diophantineapproximationandtheirrationality

1.5.1ideasintheproofthat(3)isirrational

1.5.2themeasureofirrationalityofanumber

1.5.3thethue-siegel-roththeorem,transcendental

numbers,anddiophantineequations

1.5.4proofsoftheidentities(1.5.1)and(1.5.2)

1.5.5therecurrentsequencesanandbn

1.5.6transcendentalnumbersandtheseventhhilbert

problem

1.5.7workofyu.v.nesterenkoon.e.[nes99]

2someapplicationsofelementarynumbertheory

2.1factorizationandpublickeycryptosystems

2.1.1factorizationistime-consuming

2.1.2one-wayfunctionsandpublickeyencryption

2.1.3apublickeycryptosystem

2.1.4statisticsandmassproductionofprimes

2.1.5probabilisticprimalitytests

2.1.6thediscretelogarithmproblemandthe

diffie-hellmankeyexchangeprotocol

2.1.7computingofthediscretelogarithmonelliptic

curvesoverfinitefields(ecdlp)

2.2deterministicprimalitytests

2.2.1adleman-pomerance-rumelyprimalitytest:basicideas

2.2.2gausssumsandtheiruseinprimalitytesting.

2.2.3detaileddescriptionoftheprimalitytest2.2.4primesisinp

2.2.5thealgorithmofm.agrawal,n.kayalandn.saxena

2.2.6practicalandtheoreticalprimalityproving.the

ecpp(ellipticcurveprimalityprovingbyf.morain,see[atmo93b])

2.2.7primesinarithmeticprogression

2.3factorizationoflargeintegers

2.3.1comparativedifficultyofprimalitytestingand

factorization

2.3.2factorizationandquadraticforms

2.3.3theprobabilisticalgorithmclasno

2.3.4thecontinuedfractionsmethod(cfrac)andreal

quadraticfields

2.3.5theuseofellipticcurves

partiiideasandtheories

3inductionandrecursion

3.1elementarynumbertheoryfromthepointofviewoflogic

3.1.1elementarynumbertheory

3.1.2logic

3.2diophantinesets

3.2.1enumerabilityanddiophantinesets

3.2.2diophantinenessofenumerablesets

3.2.3firstpropertiesofdiophantinesets

3.2.4diophantinenessandpell'sequation

3.2.5thegraphoftheexponentisdiophantine

3.2.6diophantinenessandbinomialcoefficients

3.2.7binomialcoefficientsasremainders

3.2.8diophantinenessofthefactorial

3.2.9factorialandeuclideandivision

3.2.10supplementaryresults

3.3partiallyrecursivefunctionsandenumerablesets

3.3.1partialfunctionsandcomputablefunctions

3.3.2thesimplefunctions

3.3.3elementaryoperationsonpartialfunctions

3.3.4partiallyrecursivedescriptionofafunction

3.3.5otherrecursivefunctions

3.3.6furtherpropertiesofrecursivefunctions

3.3.7linkwithlevelsets

3.3.8linkwithprojectionsoflevelsets

3.3.9matiyasevich'stheorem

3.3.10theexistenceofcertainbijections

3.3.11operationsonprimitivelyenumerablesets

3.3.12gsdel'sfunction

3.3.13discussionofthepropertiesofenumerablesets

3.4diophantinenessofasetandalgorithmicundecidability

3.4.1algorithmicundecidabilityandunsolvability

3.4.2sketchproofofthematiyasevichtheorem

arithmeticofalgebraicnumbers

4.1algebraicnumbers:theirrealizationsandgeometry

4.1.1adjoiningrootsofpolynomials

4.1.2galoisextensionsandfrobeniuselements

4.1.3tensorproductsoffieldsandgeometricrealizations

ofalgebraicnumbers

4.1.4units,thelogarithmicmap,andtheregulator

4.1.5latticepointsinaconvexbody

4.1:6deductionofdirichlet'stheoremfromminkowski'slemma

4.2decompositionofprimeideals,dedekinddomains,andvaluations

4.2.1primeidealsandtheuniquefactorizationproperty

4.2.2finitenessoftheclassnumber..

4.2.3decompositionofprimeidealsinextensions

4.2.4decompositionofprimesincyslotomicfields

4.2.5primeideals,valuationsandabsolutevalues

4..3localandglobalmethods

4.3.1p-adicnumbers

4.3.2applicationsofp-adicnumberstosolvingcongruenca

4.3.3thehilbertsymbol

4.3.4algebraicextensionsofqp,andthetatefield

4.3.5normalizedabsolutevalues

4.3.6placesofnumberfieldsandtheproductformula

4.3.7adelesandideles

theringofadeles

theidelegroup

4.3.8thegeometryofadelesandideles

4.4classfieldtheory

4.4.1abelianextensionsofthefieldofrationalnumbers

4.4.2frobeniusautomorphismsofnumberfieldsand

artin'sreciprocitymap

4.4.3thechebotarevdensitytheorem

4.4.4thedecompositionlawand

theartinreciprocitymap

4.4.5thekernelofthereciprocitymap

4.4.6theartinsymbol

4.4.7globalpropertiesoftheartinsymbol

4.4.8alinkbetweentheartinsymbolandlocalsymbols

4.4.9propertiesofthelocalsymbol

4.4.10anexplicitconstructionofabelianextensionsofa

localfield,andacalculationofthelocalsymbol

4.4.11abelianextensionsofnumberfields

4.5galoisgroupinarithetical'problems

4.5.1dividingacircleintonequalparts

4.5.2kummerextensionsandthepowerresiduesymbol

4.5.3galoiscohomology

4.5.4acohomologicaldefinitionofthelocalsymbol

4.5.5thebrauergroup,thereciprocitylawandthe

minkowski-hasseprinciple 5arithmeticofalgebraicvarieties 5.1arithmeticvarietiesandbasicnotionsofalgebraicgeometry

5.1.1equationsandrings

5.1.2thesetofsolutionsofasystem

5.1.3example:thelanguageofcongruences

5.1.4equivalenceofsystemsofequations

5.1.5solutionsask-algebrahomomorphisms

5.1.6thespectrumofaring

5.1.7regularfunctions

5.1.8atopologyonspec(a)

5.1.9schemes

5.1.10ring-valuedpointsofschemes

5.1.11solutionstoequationsandpointsofschemes

5.1.12chevalley'stheorem

5.1.13somegeometricnotions

5.2geometricnotionsinthestudyofdiophantineequations

5.2.1basicquestions

5.2.2geometricclassification

5.2.3existenceofrationalpointsandobstructionstothe

hasseprinciple

5.2.4finiteandinfinitesetsofsolutions

5.2.5numberofpointsofboundedheight

5.2.6heightandarakelovgeometry

5.3ellipticcurves,abelianvarieties,andlineargroups

5.3.1algebraiccurvesandriemannsurfaces

5.3.2ellipticcurves

5.3.3tarecurveanditspointsoffiniteorder

5.3.4themordell-weiltheoremandgaloiscohomology

5.3.5abelianvarietiesandjacobians

5.3.6thejacobianofanalgebraiccurve

5.3.7siegel'sformulaandtamagawa'measure

5.4diophantineequationsandgaloisrepresentations

5.4.1thetaremoduleofanellipticcurve

5.4.2thetheoryofcomplexmultiplication

5.4.3charactersofl-adicrepresentations

5.4.4representationsinpositivecharacteristic

5.4.5thetatemoduleofanumberfield

5.5thetheoremoffairingsandfinitenessproblemsindiophantinegeometry

5.5.1reductionofthemordellconjecturetothefinitenessconjecture

5.5.2thetheoremofshafarevichonfinitenessforellipticcurves

5.5.3passagetoabelianvarieties

5.5.4finitenessproblemsandtatesconjecture

5.5.5reductionoftheconjecturesoftatetothefinitenesspropertiesorisogenies5.5.6thefaltings-arakelovheight

5.5.7heightsunderisogeniesandconjecturetzetafunctionsandmodularforms

6.1zetafunctionsofarithmeticschemes

6.1.1zetafunctionsofarithmeticschemes

6.1.2analyticcontinuationofthezetafunctions

6.1.3schemesoverfinitefieldsanddeligne'stheorem

6.1.4zetafunctionsandexponentialsums

6.2l-functions,thetheoryoftateandexpliciteformulae

6.2.1l-functionsofrationalgaloisrepresentations

6.2.2theformalismofartin

6.2.3example:thededekindzetafunction

6.2.4heckecharactersandthetheoryoftare

6.2.5explicitformulae

6.2.6theweilgroupanditsrepresentations

6.2.7zetafunctions,l-functionsandmotives

6.3modularformsandeulerproducts

6.3.1alinkbetweenalgebraicvarietiesandl-functions

6.3.2classicalmodularforms

6.3.3application:tatecurveandsemistableellipticcurves

6.3.4analyticfamiliesofellipticcurvesandcongruence

subgroups

6.3.5modularformsforcongruencesubgroups

6.3.6hecketheory

6.3.7primitiveforms

6.3.8weil'sinversetheorem

6,4modularformsandgaloisrepresentations

6.4.1ramanujan'scongruenceandgaloisrepresentations

6.4.2alinkwitheichler-shimura'sconstruction

6.4.3theshimura-taniyama-weilconjecture

6.4.4theconjectureofbirchandswinnerton-dyer

6.4.5theartinconjectureandcuspforms

theartinconductor

6.4.6modularrepresentationsoverfinitefields

6.5automorphicformsandthelanglandsprogram

6.5.1arelationbetweenclassicalmodularformsand

representationtheory

6.5.2automorphicl-functions

furtheranalyticpropertiesofautomorphicl-functions

6.5.3thelanglandsfunctorialityprinciple

6.5.4automorphicformsandlanglandsconjectures

fermat'slasttheoremandfamiliesofmodularforms

7.1shimura-taniyama-weilconjectureandreciprocitylaws

7.1.1problemofpierredefermat(1601-1665)

7.1.2g.lam6'smistake

7.1.3ashortoverviewofwiles'marvelousproof

7.1.4thestwconjecture

7.1.5aconnectionwiththequadraticreciprocitylaw

7.1.6acompleteproofofthestwconjecture

7.1.7modularityofsemistableellipticcurves

7.1.8structureoftheproofoftheorem7.13(semistable

stwconjecture)

7.2theoremoflanglands-tunnelland

modularitymodulo3

7.2.1galoisrepresentations:preparation

7.2.2modularitymodulop

7.2.3passagefromcuspformsofweightonetocuspforms

ofweighttwo..

7.2.4preliminaryreviewofthestagesoftheproofoftheorem7.13onmodularity

7.3modularityofgaloisrepresentationsanduniversaldeformationrings

7.3.1galoisrepresentationsoverlocalnoetherianalgebras

7.3.2deformationsofgaloisrepresentations

7.3.3modulargaloisrepresentations

7.3.4admissibledeformationsandmodulardeformations

7.3.5universaldeformationrings

7.4wiles'maintheoremandisomorphismcriteriaforlocalrings

7.4.1strategyoftheproofofthemaintheorem7.33

7.4.2surjectivityof

7.4.3constructionsoftheuniversaldeformationring

7.4.4asketchofaconstructionoftheuniversalmodulardeformationringt2

7.4.5universalityandthechebotarevdensitytheorem

7.4.6isomorphismcriteriaforlocalrings

7.4.7j-structuresandthesecondcriterionofisomorphismoflocalrings

7.5wiles'inductionstep:applicationofthecriteriaandgaloiscohomology

7.5.1wiles'inductionstepintheproofofmaintheorem7.33

7.5.2aformularelatingpreparation

7.5.3theselmergroupand

7.5.4infinitesimaldeformations

7.5.5deformationsoftype

7.6therelativeinvariant,themaininequalityandtheminimalcase

7.6.1therelativeinvariant

7.6.2themaininequality

7.6.3theminimalcase

7.7endofwiles'proofandtheoremonabsoluteirreducibility

7.7.1theoremonabsoluteirreducibility

7.7.2fromp=3top=5

7.7.3familiesofellipticcurveswithfixede

7.7.4theendoftheproofthemostimportantinsightspartiiianalogiesandvisionsiii-0introductorysurveytopartiii:motivationsanddescription

iii.1analogiesanddifferencesbetweennumbersandfunctions:point,archimedeanpropertiesetc

iii.1.1cauchyresidueformulaandtheproductformula

iii.l.2arithmeticvarieties

iii.l.3infinitesimalneighborhoodsoffibers

iii,.2arakelovgeometry,fiberovercycles,greenfunctions

(d'apresgillet-soule)

iii.2.1arithmeticchowgroups

iii.2.2arithmeticintersectiontheoryandarithmetic

riemann-rochtheorem

iii.2.3geometricdescriptionoftheclosedfibersatinfinity

iii.3-functions,localfactorsatserre'sf-factors

iii.3.1archimedeanl-factors

iii.3.2deninger'sformulae

iii.4aguessthatthemissinggeometricobjectsare

noncommutativespaces

iii.4.1typesandexamplesofnoncommutativespaces,and

howtoworkwiththem.noncommutativegeometry

andarithmetic

isomorphismofnoncommutativespacesandmoritaequivalence

thetoolsofnoncommutativegeometry

iii.4.2generalitiesonspectraltriples

iii.4.3contentsofpartiii:descriptionofpartsofthisprogram

8arakelovgeometryandnoncommutativegeometry

8.1schottkyuniformizationandarakelovgeometry

8.1.1motivationsandthecontextoftheworkof

consani-marcolli

8.1.2analyticconstructionofdegeneratingcurvesover

completelocalfields

8.1.3schottkygroupsandnewperspectivesinarakelov

geometry

schottkyuniformizationandschottkygroups

fuchsianandschottkyuniformization

8.1.4hyperbolichandlebodies

geodesicsinr

8.1.5arakelovgeometryandhyperbolicgeometry

arakelovgreenfunction

crossratioandgeodesics

differentialsandschottkyuniformization

greenfunctionandgeodesics

8.2cohomologicalconstructions

8.2.1archimedeancohomology

operators

sl(2,r)representatious

8.2.2localfactorandarchimedeancohomology

8.2.3cohomologicalconstructions

8.2.4zetafunctionofthespecialfiberandreidemeister

torsion

8.3spectraltriples,dynamicsandzetafunctions

8.3.1adynamicaltheoryatinfinity

8.3.2homotopyquotion

8.3.3filtration

8.3.4hilbertspaceandgrading

8.3.5cuntz-kriegeralgebra

spectraltriplesforschottkygroups

8.3.6arithmeticsurfaces:homologyandcohomology

8.3.7archimedeanfactorsfromdynamics

8.3.8adynamicaltheoryformumfordcurves

genustwoexample

8.3.9cohomologyof

8.3.10spectraltriplesandmumfordcurves

8.4reductionmod

8.4.1homotopyquotientsand"reductionmodinfinity"

8.4.2baum-connesmap

references

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