椭圆曲线 数学计算机的应用

ELLIPTICCURVES

J.S.MILNE

August21,1996;vl.Ol

ABSTRACT.ThesearethenotesforMath679,UniversityofMichigan.Winter1996,exactly

astheywerehandedoutduringthecourseexceptforsomeminorcorrections.

Pleasesendcommentsandcorrectionstomeatjmilne@using“Math679“as

thesubject.

CONTENTS

Introduction1

Fastfactorizationofintegers

Congruentnumbers

Fermat5slasttheorem

1.ReviewofPlaneCurves2

Affineplanecurves

Projectiveplanecurves

2.RationalPointsonPlaneCurves.6

Hensel'slemma.

Abriefintroductiontothep-adicnumbers

Somehistory

3.TheGroupLawonaCubicCurve12

4.FunctionsonAlgebraicCurvesandtheRiemann-RochTheorem14

Regularfunctionsonaffinecurves

Regularfunctionsonprojectivecurves

TheRiemann-Rochtheorem

Thegrouplawrevisited

Perfectbasefields

5.DefinitionofanEllipticCurve19

Planeprojectivecubiccurveswitharationalinflectionpoint

Generalplaneprojectivecurves

Completenonsingularcurvesofgenus1

Copyright1996J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.

i

J.S.MILNE

Thecanonicalformoftheequation

Thegrouplawforthecanonicalform

ReductionofanEllipticCurveModulop23

Algebraicgroupsofdimension1

Singularcubiccurves

Reductionofanellipticcurve

Semistablereduction

Reductionmodulo2and3

Otherfields

7.EllipticCurvesover29

8.TorsionPoints32

Formulas

SolutiontoExercise4.8

9.NeronModels37

Weierstrassminimalmodels

TheworkofKodaira

ThecompleteNeronmodel

Summary

10.EllipticCurvesovertheComplexNumbers41

Latticesandbases

QuotientsofCbylattices

Doublyperiodicfunctions

TheholomorphicmapsC/A—C/Az

TheWeierstrasspfunction

Eisensteinseries

Thefieldofdoublyperiodicfunctions

TheellipticcurveF(A)

ClassificationofellipticcurvesoverC

Torsionpoints

Endomorphisms

Appendix:Resultants

11.TheMordell-WeilTheorem:StatementandStrategy54

12.Groupcohomology55

Cohomologyoffinitegroups

CohomologyofinfiniteGaloisgroups

13.TheSelmerandTate-Shafarevichgroups59

ELLIPTICCURVESiii

14.TheFinitenessoftheSelmerGroup60

ProofofthefinitenessoftheSelmergroupinaspecialcase

ProofofthefinitenessoftheSelmergroupinthegeneralcase

15.Heights65

HeightsonP1

HeightsonE

16.CompletionoftheProofoftheMordell-WeilTheorem,andFurtherRe­

marks70

TheProblemofComputingtheRankofE(Q)

TheNeron-TatePairing

Computingtherank

17.GeometricInterpretationoftheCohomologyGroups;Jacobians75

Principalhomogeneousspaces(ofsets)

Principalhomogeneousspaces(ofcurves)

Theclassificationofprincipalhomogeneousspaces

GeometricInterpretationof

GeometricInterpretationoftheExactSequence

TwistsofEllipticCurves

Curvesofgenus1

TheclassificationofellipticcurvesoverQ(summary)

18.TheTate-ShafarevichGroup;FailureOfTheHassePrinciple83

19.EllipticCurvesOverFiniteFields86

TheFrobeniusmap;curvesofgenus1overFp

Zetafunctionsofnumberfields

Zetafunctionsofaffinecurvesoverfinitefields

ExpressionofZ(C,T)intermsofthepointsofC

Zetafunctionsofplaneprojectivecurves

Therationalityofthezetafunctionofanellipticcurve

ProofoftheRiemannhypothesisforellipticcurves

ABriefHistoryofZeta

20.TheConjectureofBirchandSwinnerton-Dyer100

Introduction

ThezetafunctionofavarietyoverQ

ThezetafunctionofanellipticcurveoverQ

StatementoftheConjectureofBirchandSwinnerton-Dyer

What'sknownabout,theconjectureofB-S/D

ivJ.S.MILNE

21.EllipticCurvesandSpherePackings106

Spherepackings

Example

22.AlgorithmsforEllipticCurves110

23.TheRiemannSurfacesX()(N)112

ThenotionofaRiemannsurface

QuotientsofRiemannsurfacesbygroupactions

TheRiemannsurfacesX(「)

Thetopologyonr\IHI*

ThecomplexstructureonFo(N)\H*

ThegenusofX°(N)

24.Xo(N)asanAlgebraicCurveoverQ119

Modularfunctions

ThemeromorphicfunctionsonX。⑴

ThemeromorphicfunctionsonXo(N)

ThecurveX()(N)overQ

ThepointsonthecurveX()(N)

Variants

25.ModularForms125

Definitionofamodularform

Themodularformsforr()⑴

26.ModularFormsandtheL-seriesofEllipticCurves128

DirichletSeries

TheL-seriesofanellipticcurve

L-seriesandisogenyclasses

TheL-seriesofamodularform

ModularformswhoseL-serieshaveafunctionalequations

ModularformswhoseL-functionsareEulerproducts

DefinitionoftheHeckeoperators

Linearalgebra:thespectraltheorem

ThePeterssoninnerproduct

Newforms:thetheoremofAtkinandLehner

27.StatementoftheMainTheorems140

28.HowtogetanEllipticCurvefromaCuspForm142

DifferentialsonRiemannsurfaces

TheJacobianvarietyofaRiemannsurface

ConstructionoftheellipticcurveoverC

ELLIPTICCURVESv

ConstructionoftheellipticcurveoverQ

29.WhytheL-SeriesofEAgreeswiththeL-Seriesoff147

Theringofcorrespondencesofacurve

TheHeckecorrespondence

TheFrobeniusmap

Briefreviewofthepointsoforderponellipticcurves

TheEichler-Shimurarelation

Thezetafunctionofanellipticcurverevisited

TheactionoftheHeckeoperatorsonZ)

Theproofthatc(p)=ap

30.Wiles'sProof153

31.Fermat,AtLast156

Bibliography157

ELLIPTICCURVES1

INTRODUCTION

Anellipticcurveoverafieldkisanonsingularcompletecurveofgenus1withadistin­

guishedpoint.Ifcharfc*2,3,itcanberealizedasaplaneprojectivecurve

222

yZ=X3+aXZ+bZ\4Q3+276*0,

andeverysuchequationdefinesanellipticcurveoverk.Asweshallsee,thearithmetic

theoryofellipticcurvesoverQ(andotheralgebraicnumberfields)isarichabeautiful

subject.Manyimportantphenomenafirstbecomevisibleinthestudyellipticcurves,and

ellipticcurveshavebeenusedsolvesomeveryfamousproblemsthat,atfirstsight,appear

tohavenothingtodowithellipticcurves.Imentionthreesuchproblems.

Fastfactorizationofintegers.Thereisanalgorithmforfactoringintegersthatuses

ellipticcurvesandisinmanyrespectsbetterthanpreviousalgorithms.See[K2,VI.4],

[ST,IV.4],or[C2,Chapter26].Peoplehavebeenfactoringintegersforcenturies,butrecently

thetopichasbecomeofpracticalsignificance:givenanintegernwhichistheproductn=pq

oftwo(large)primespandq,thereisacodeforwhichanyonewhoknowsncanencodea

message,butonlythosewhoknowp,qcandecodeit.Thesecurityofthecodedependson

nounauthorizedpersonbeingabletofactorn.

Congruentnumbers.Anaturalnumbernissaidtobecongruentifitoccursasthearea

ofarighttrianglewhosesideshaverationallength.Ifwedenotethelengthsofthesidesby

%n,thennwillbecongruentifandonlyiftheequations

9701

x+y=z、n=^xy

havesimultaneoussolutionsinQ.TheproblemwasofinteresttotheGreeks,andwas

discussedsystematicallybyArabscholarsinthetenthcentury.Fibonaccishowedthat5and

6arecongruent,Fermatthat1,2,3,arenotcongruent,andEulerprovedthat7iscongruent,

buttheproblemappearedhopelessuntilin1983Tunnellrelatedittoellipticcurves.

Fermafslasttheorem.RecentlyWilesprovedthatallellipticcurvesoverQ(withamild

restriction)ariseinacertainfashionfrommodularforms.Itfollowsfromhistheorem,that

foranoddprimep#3,theredoesnotexistanellipticcurveoverQwhoseequationhasthe

form

y2=X(X+Q)(X-b)

witha,b,a+ballpowersofintegers,i.e.,theredoesnotexistanontrivialsolutioninZ

totheequation

X。+Yp=Z。；

—Fermat^LastTheoremisproved!

Thecoursewillbeanintroductorysurveyofthesubjectoftenproofswillonlybe

sketched,butIwilltrytogiveprecisereferencesforeverything.

Therearemanyexcellentbooksonsubject-seetheBibliography.Silverman[S1,S2]is

becomingthestandardreference.

2J.S.MILNE

1.REVIEWOFPLANECURVES

Affineplanecurves.Letkbeafield.TheaffineplaneoverkisA2(fc)=k'2.

Anonconstantpolynomialfek[X,V],assumedtohavenorepeatedfactorinka][X,Y],

definesaplaneaffinecurveCfoverkwhosepointswithcoordinatesinanyfieldKDkare

thezerosoffinK2:

Cf(K)={(©y)EK2|F(x,y)=0}.

ThecurveCissaidtobeirreducibleiffisirreducible,anditissaidbegeometrically

irreducibleiffremainsirreducibleoverfcal(equivalent^,overanyalgebraicallyclosedfield

containingAr).

Sincek[X,Y]isauniquefactorizationdomain,wecanwriteanyfasaboveasaproduct

f=/1/2,,•/r°fdistinctirreduciblepolynomials,andthen

Cf=Cf}U・・・UC7r

withtheC//.irreduciblecurves.TheCy.arecalledtheirreduciblecomponentsofC/.

Example1.1.(a)Let/《X,Y)beanirreduciblepolynomialinY],noconstant

multipleofwhichliesQ[X,Y],andlet/i(X,Y)beitsconjugateoverQ(i.e.,replaceeach

，2with—，2).Then/(X,Y)=可/i(X,Y)&(X,Y)liesinQ[X,Y]becauseitisfixedby

theGaloisgroupofQ[，2]/Q.ThecurveCfisirreduciblebutnotgeometricallyirreducible.

(b)Letkbeafieldofcharacteristicp.Assumekisnotperfect,sothatthereexistsan

aek,akp.Consider

f(X,Y)=Xp+aYp.

Thenfisirreducibleink[X,F],butinfcal[X,Y]itequals(X+aY)pwhereo?=a(remember,

thebinomialtheoremtakesonaspeciallysimpleformforpthpowersincharacteristicp).

Thusfdoesnotdefineacurve.

Wedefinethepartialderivativesofapolynomialbytheobviousformulas.

LetP—(a,6)GC/(X),someKDk.Ifatleastoneofthepartialderivatives£,靠is

nonzeroatP,thenPissaidtobenonsingular,andthetangentlinetoCatPis

(&JX-W(i=0.

AcurveCissaidtobenonsingularifallthepointsinarenonsingular.Acurveor

pointthatisnotnonsingularsaidtobesingular.

Aside1.2.Lety)beareal-valuedfunctiononR2.InMath215onelearnsthatV/=可

(K，fy)isavectorfieldonR2that,atanypointP=(a,b)GR2,pointsinthedirection

inwhichJ(x,y)increasesmostrapidly(i.e.,hasthemostpositivedirectionalderivative).

Hence(V/)pisnormaltoanylevelcurve/(T,y)=cthroughF,andtheline

(▽/)P・(X—Q,Y—b)=0

passesthroughPandisnormaltothenormaltothelevelcurve.Itisthereforethetangent

line.

ELLIPTICCURVES3

Example1.3.Considerthecurve

c：产=+Q.X+b.

AtasingularpointofC

2Y=0,3X?+Q=0,产=+ax+b.

Assumechark*2.HenceY=QandXisacommonrootofX3+aX+bandits

derivative,i.e.,adoublerootofX3+aX+b.ThusCisnonsingular4=>X3+aX+bhas

nomultipleroot(infcal)<=>itsdiscriminant4a3+27/isnonzero.

Assumechark=2.ThenCalwayshasasingularpoint(possiblyinsomeextensionfield

ofk),namely,wherea：2+a=0and俨—+aa+6.

LetP=(a,b)€Cf(K).Wecanwrite/asapolynomialmX—aandY—bwith

coefficientsinK、say,

".)=/i(x_a,y_4+・・・+/n(x__b)

whereishomogeneousofdegreeiinX—aandY—h(thistheTaylorexpansionoffl).

ThepointPisnonsingularifandonlyiffi*0,inwhichcasethetangentlinetoC/atP

hasequation/i=0.

SupposethatPissingular,sothat

/(X,F)=fm(X—a,Y—b)+termsofhigherdegree,

wherefm7^0,m>2.ThenPissaidtohavemultiplicitymonC,denotedmp(C).If

m=2,thenPiscalledadoublepoint.Forsimplicity,take(a,b)=(0,0).Then(over

♦(X,y)=nw

whereeachLiisahomogeneouspolynomialQX+&YofdegreeonewithcoefficientsinA：al.

ThelinesLi=0arecalledthetangentlinestoC(atP,and%iscalledmultiplicityofL,.

ThepointPissaidtobeanordinarysingularityifthetangentlinesarealldistinct,i.e.,

ri=1foralli.Anordinarydoublepointiscalledanode.

Example1.4.ThecurveY2=X,3+aX2hasasingularityat(0,0).Ifa*0,itisanode,

andthetangentlinesat(0,0)areY=±，QX.Theyaredefinedoverkifandonlyifaisa

squareink.

IfQ=0,thesingularityisacusp.(AdoublepointPonacurveCiscalledacuspifthere

isonlyonetangentlineLto(7atP,and,withthenotationdefinedbelow,/(P,LAC)=3.)

2

ConsidertwocurvesCfandCginA(fe),andletPeC*/(X)Dg(K),someKDk.

AssumethatPisanisolatedpointofCfACg,i.e.,CfandCgdonothaveacommon

irreduciblecomponentpassingthroughP.WedefinetheintersectionnumberofCyandCg

atPtobe

KP,CfQCg)=dimKK[X,¥]—/(/,g)

(dimensionasX-vectorspaces).

Remark1.5.IfCfandCghavenocommoncomponent,then

EnQ)=din.k[X,Y]/U,g).

尸€C(A：a】)CC(ka】)

ThisisparticularlyusefulwhenCfandCgintersectatasinglepoint.

4J.S.MILNE

Example1.6.LetCbethecurveY2=X3,andletL:Y=0beitstangentlineat

P=(0,0).Then

233

I(F,£nC)=dimkk[X,Y]/(Y.Y-X)=dimfck[X]/(X)=3.

Remark1.7.(a)Theintersectionnumberdoesn'tdependonwhichfieldKthecoordinates

ofPareconsideredtoliein.

(b)Asexpected,/(F,CnZ?)=1ifandonlyifPisnonsingularonbothCandD,andthe

tangentlinestoCandDatParedistinct.Moregenerally,/(P,CC\D)>

withequalityifandonlyifCandDhavenotangentlineincommonatP.

Projectiveplanecurves.Theprojectiveplaneoverkis

呼⑻={(a;,y,z)ek3\(x,y,z)*(0,0,0)}/〜

where(①，y,z)〜(/,y1ifandonlyifthereexistsac#0suchthaty\z')=(CN,cy,cz).

Wewrite(x:y:z)fortheequivalenceclass1of(①，y,z).LetPGP2(fc);thetriples(①，y,z)

representingPlieonasinglelineL(P)throughtheoriginink3,andP一L(P)isabijection

fromP2(fc)tothesetofallsuchlines.

Projectiven-spacePn(fc)canbedefinedsimilarlyforanyn>0.

LetUQ={(z:g：z}|z00},andlet={(z:y:z)\z=0}.Then

(z,0)i(c：y：1)：A2(fc)—UQ

isabijection,and

(s:y)i(?:y:0):W/)-Lg(k)

isabijection.Moreover,P2(A;)isthedisjointunion

ofthe"affineplane"UQwiththe“lineatinfinity"L^.Aline

aX+bY+cZ=Q

meetsatthepoint(—b:Q:0)=(1:—。0).ThuswecanthinkofP2(fe)asbeingthe

affineplanewithexactlyonepointaddedforeachfamilyofparallellines.

AnonconstanthomogeneouspolynomialFGassumedtohavenorepeated

factorink%definesaprojectiveplanecurveCFoverkwhosepointsinanyfieldKDkare

thezerosofFinP2(K):

。尸(K)={(x:y:z)\F(x,yYz)=0}.

Notethat,becauseFishomogeneous,

F(cx,cy,cz)=cde&FF(x,y,z),

andso,althoughitdoesn'tmakesensetospeakofthevalueofFatapointofP2,itdoes

makesensetosaywhetherornotFiszeroatP.Again,thedegreeofFiscalledthe

degreeofthecurveC,andaplaneprojectivecurveis(uniquely)aunionofirreducibleplane

projectivecurves.

Thecurve

y2Z=X3+aXZ2+bZ3

iThecolonismeanttosuggestthatonlytheratiosmatter.

ELLIPTICCURVES5

intersectsthelineatinfinityatthepoint(0:1:0),i.e.,atthesamepointasallthevertical

linesdo.Thisisplausiblegeometrically,because,asyougoouttheaffinecurve

Y2^X3+aX+h

withincreasingxandy,theslopeofthetangentlinetendstooo.

LetUi={(/:y:z}\y0},andletU2={(n:y:z)\x*0}.ThenUiandU2areagain,

inanaturalway,affineplanes;forexample,wecanidentify«withA2(fc)via

(7:1:2)一(x,z).

Sinceatleastoneofn,y,orzisnonzero,

/⑻=UoUUiUU2.

AplaneprojectivecurveC=CFistheunionofthreecurves,

C=CoUClUC25Ci=CnUi.

WhenweidentifyeachUiwithA2(fc)inthenaturalway,thenandC?become

identifiedwiththeaffinecurvesdefinedbythepolynomialsF(X,K,1),F(X,1,Z),and

F(l,y,Z)respectively.

Thecurve

C:y2z=X3+aX#+bZ3

isunusual,inthatitiscoveredbytwo(ratherthan3)affinecurves

Co：Y2=X3+aX+b

and

Ci:Z=X3+QXZ?+bZ3.

Thenotionsoftangentline,multiplicity,etc.canbeextendedtoprojectivecurvesby

notingthateachpointPofaprojectivecurveCwilllieonatleastoneoftheaffinecurves

Exercise1.8.LetPbeapointonaplaneprojectivecurveC=CF，ShowthatPis

singular,i.e.,itissingularontheplaneaffinecurveGforone(henceall)iifandonlyif

F(P)=0=(fy)=(器)=(歌).IfPisnonsingular,showthattheplaneprojective

line

Z=0

心黑）产得力图p

hasthepropertythatLAisthetangentlinetotheaffinecurveCCU&forz=0,1,2.

Theorem1.9(Bezout).LetCandDbeplaneprojectiveofdegreesmandnrespectively

overk,andassumethattheyhavenoirreduciblecomponentincommon.Thentheyintersect

al

overkinexactlymnpoints,countingmultiplicitiesfie,

£Z(P,Cn/?)=mn.

Proof.See[F]pll2,ormanyotherbooks.

6J.S.MILNE

Forexample,acurveofdegreemwillmeetthelineatinfinityinexactlympoints,counting

multiplicities.Ourfavouritecurve

C:y2Z=X3+aXZ2+bZ3

meetsLgatasinglepointP=(0:1:0),butI(P,LgClC)=3.[Exercise:Provethis!]In

general,anonsingularpointPonacurveCiscalledapointofinflectioniftheintersection

multiplicityofthetangentlineand(7atPis>3.

Supposekisperfect.ThenallthepointsofC(fcal)Cl。(炉])willhavecoordinatesinsome

finiteGaloisextensionKofk,andtheset

C(K)AO(K)CP2(7<)

isstableundertheactionofGal(K/k).

Remark1.10.(Fortheexperts.)Essentially,wehavedefinedanaffine(resp.projective)

curvetobeageometricallyreducedclosedsubschemeofA差(resp.吸)ofdimension1.Such

aschemecorrespondstoanidealofheightone,whichisprincipal,becausepolynomialrings

areuniquefactorizationdomains.Thepolynomialgeneratingtheidealoftheschemeis

uniquelydeterminedbytheschemeuptomultiplicationbyanonzeroconstant.Theother

definitionsinthissectionarestandard.

References:Thebestreferenceforwhatlittleweneedfromalgebraicgeometryis[F].

2.RATIONALPOINTSONPLANECURVES.

LetCbeaplaneprojectivecurveoverQ(orsomeotherfieldwithaninterestingarith­

metic),definedbyahomogeneouspolynomialF(X,K,Z),Thetwofundamentalquestions

indiophantinegeometrythenare:

Question2.1.(a)DoesChaveapointinQ,thatis,doesF(X,Y,Z)haveanontrivialzero

inQ?

(b)Iftheanswerto(a)isyes,canwedescribethesetofcommonzeros?

Thereisalsothequestionofwhetherthereisanalgorithmtoanswerthesequestions.

Forexample,wemayknowthatacurvehasonlyfinitelymanypointswithouthavingan

algorithmtoactuallyfindthepoints.

Forsimplicity,intheremainderofthissection,I'llassumethatCisabsolutelyirreducible,

i.e.,thatF(X,Y,Z)isirreducibleoverQal.

Hereisoneobservationthatweshallusefrequently.LetKbeafinite(ofeveninfinite)

GaloisextensionofQ,andlet

/(x,y)=£%xkwQ[x'.

If(a,b)6K2isazeroof以X,K),thensoalsois(OQ,ob)foranyoGGal(K/Q),because

0=(Q,b)=O(£Q〃Q%)=£049。)'(4加=

ThusGal(K/Q)actsonC(K),whereCistheaffinecurvedefinedbyMore

generally,ifCi,C2,...areaffinecurvesoverQ,thenGal(K/Q)stabilizesthesetC\(K)A

02(K)….Onapplyingthisremarktothecurves/(X,Y)=0,聂(X,Y)=0,柒(X)Y)=0,

weseethatGal(K/Q)stabilizesthesetofsingularpointsinSimilarremarksapply

toprojectivecurves.

ELLIPTICCURVES7

Curvesofdegreeone.Firstconsideracurveofdegreeone,i.e.,aline,

C:aX+hY+cZ=Q,a,h,cinQandnotallzero.

Italwayshaspoints,anditispossibletoparameterizethepoints:if,forexample,。羊0,the

map

/、/ab、

(s:力)1(s:力：一s—t)

cc

isabijectionfromPx(fc)ontoC(fe).

Curvesofdegreetwo.InthiscaseF(X,Y,Z)isaquadraticformin3variables,andCisa

conic.NotethatCcan'tbesingular:ifPhasmultiplicitym,then(accordingto(1.7b))a

lineLthroughPandasecondpointQonthecurvewillhave

/(RLAC)+I(Q,LnC)>2+1=3,

whichviolatesBezout'stheorem.

SometimesitiseasytoseethatC(Q)=0.Forexample,

X2+Y2+Z2

hasnonontrivialzerobecauseithasnonontrivialrealzero.Similarly,

X'2+Y2-3Z2

hasnonontrivialzero,becauseifitdiditwouldhaveazero(x,y,z)withx,y,z6Zand

gcd®y,z)=1.TheonlysquaresinZ/3Zare0and1,andso

x2+y2=0mod3

impliesthatx=Q=ymod3.Butthen3mustdivide%,whichcontradictsourassumption

thatgcd(%y、z)=1.Thisargumentshows,infact,thatX'2+Y'2—3Z2doesnothavea

nontrivialzerointhefieldQ3of3-adicnumbers.

Theseexamplesillustratetheusefulnessofthefollowingstatement:anecessarycondition

forCtohaveapointwithcoordinatesinQisthatithaveapointwithcoordinatesinR

andinQpforallp.AtheoremofLegendresaysthattheconditionisalsosufficient:

Theorem2.2(Legendre).AquadraticformF(X,KZ)withcoefficientsinQhasanon­

trivialzeroinQifandonlyifithasanontrivialzeroinIRandinforallp.

Remark2.3.(a)ThisisnotquitehowLegendre(17521833)statedit,sincep-adicnumbers

arelessthan100yearsold

(b)Thetheoremdoesinfactgiveapracticalalgorithmforshowingthataquadraticform

doeshaveanontrivialrationalzero——see(2.11)below.

(c)ThetheoremistrueforquadraticformsF(X(),,Xn)inanynumberofvariables

overanynumberfieldK(Hasse-Minkowskitheorem).Thereisaverydown-to-earthproof

oftheoriginalcaseofthetheoremin[C2]ittakesthreelectures.Agoodexpositionof

theproofforformsoverQinanynumberofvariablesistobefoundinSerre,Courseon

Arithmetic.Thekeycasesare3and4variables(2istrivial,andfor>5variables,oneuses

inductiononn),andthekeyresultneededforitsproofisthequadraticreciprocitylaw.

FornumberfieldsKotherQ,theproofrequirestheHilbertreciprocitylaw,whichisbest

derive