Table Of Content215
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribcl
Springer
New York
Berlin
Heidelberg
Hong Kong
London
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Graduate Texts in Mathematics
I TAKEVTIUARING. Introduclion lo SP~.ERP.r inciples of Random Walk.
Axiomatic Sa Theory. 2nd cd. 2nd nl.
2 Orronv. Mcuure and Category. 2nd ed. ALEUNDFR~VERMFSRe. veral Complex
3 SCIIAF~R. I'opological Vector Spaces. Variahlcs and Ijannch Algebras. 3rd cd.
2nd ctl. KCI.I.EYMAMIOKA ct al. 1-tncar
4 II ILTON/STAM~A~AH C. ourse in Topological Spaccs.
Ilomok>gicalA lgchra. 2nd ed. MONK.M athcnurical Logic.
5 MACI AVC.at cpories for lhc Working C;utrrn~il:~~nsc-Sme vml Complex
Mathematician. 2nd cd. Vanahlcs.
6 II I:C;IIPS/~'IPCR. Prcjcctive Plancs. ARVFSONA.n Invitation lo C-Algebras.
7 J.4' SFRRCA. Course in Arithmetic. K~:~rrwlSm&~~rIk.n umcr~ble
R TAKEIT~~//ARA~.X IO ~ICS~~TChc oty hlarkov Chains. 2nd ed.
9 ~IIIMPIInItRrodFucYtionS to .L ie Algehm~ APOSTOLM. ndular Functions and
and IZcpresentation Thcory. Dirichlcl Scries in Numhcr Thcory.
10 COIIFN. A Course in Simple Homotopy 2nd cd.
111cory. J.-P. SERREL. inear Representations of
II CONWAYF.u nctions ofO ne Complex Finite Groups.
Variable I.2 nd en. ~ILLMAN/J~~RISORNin.s s of Continuous
12 RFALS. Advanced Mathemalical Analysis. Functions.
I3 ASDER~OS~~:I!I.IFR. Rings and Categorici KFICDIG.E kmentary Algebmic <.mrmtq.
of Xl~dtilcs2. nd ed. I.ntv~P..r ohahility 'Theory I.4 th ed.
14 ~;o~.rrtctrsrv/(;rnu.~S~tmab.l e Mappings 1~)CwP. rohah~l~Ithyc ory 11. 4th ed.
and Ihcir Singularttics. MOISE.( ieomctnc Topology in
15 I~FRRERIAINc .c tures in Functional Ihmcnsions 2 and 3.
Analysts and Opmlor b r y . SACHSNLC~m. wnl Relativity for
16 WISTFR'.n w Structure of 11elds. Matl~ema~icians.
17 ROSIIN~LATTK. andon~P rocnscs. 2nd ed. <;RUI:NHERGIWEIR.L inear Geometry.
I8 II AI.MMOcasSurc. l hcory. 2nd cd.
10 Il~t.scosh. lhlhcrt Spacc Problem Rook. Ena~ntsF.e rnrat's Last Thcom.
2nd cd. KI.IN<~FN~ERGA Course in DiflmtiaI
20 Iltrs~h~ot.F~th.r~e ~D.u ndles. 3rd ed. (imnrlry.
2 1 111I ~IPIIRI:YS. I. incar Alpchraic Groups. ~IARTSAlgIcIhOmicR GNeomEetr.y .
22 I~ARNI:S~MACKA. n Alpchnic lntrocluction MANIS, A Counc In Mathemat~caLl ogic.
to hlathcmat~calln gtc. GIUVER'WATKINS.C ornb~natoricsw ith
23 (;RI.IIR. Llnc~Ar lpchra. 4th ed. Emphasis on the Thcory of Graphs.
24 11ot.hi~sG. comctrtc I:unctional Analysis I~ROWNIPFARICntYro. duction 10 Operator
and Its Appllcat~ons. 'lhcory I: Elements of 1:unctional Analysis
25 IlrwrniS~uortnr~Rcie.a l and Abstract MASSEYA.l gchraic Topolo~yA. n
Analysis. Introductton.
26 MANESA. lgchmic Theories. ('~ow~t.~JFoInxtr.o duction to Knot
27 KFI.I.I:Y. Gcncnl Topolopy. Ihcory
?I ZARISRI~SAS(I+<..' omniutalive Algchra. Kon1.m.. p-adic Numbers. p-adic
Vl1l.l Analys~sa, nd Zeta-Functions. 2nd ed.
2') ~ARISKI/SAMLIEI..C ommutntivc Algehn. IA NG. Cyclotom~cF ~clds.
Vol 11. ARNOLDh. lrlhcmatical Mclhnds in
30 JA(Y~HSOLeNc.t ures In Ahstracl Algchtn I. Clas~~cMaCl C~PIIICS.2 nd cd.
Ihvc ConccpLr. H'tln I~I:AI>. 1':lcmcnts of Il omolopy
31 J~c~onsnNlm. turcs in Ahstract Algebra !I. Ihcory.
1.1ncar Algchra. KARGA~OLOV/M~R~1JuAnKdaOm~c'n.t als
32 J~coesosI rclurcs in Ahsrnct Algchn of the Theory of Groups.
Ill lhruiry nfI:~cldsa nd (ialo~sT licory. 1101.1nn~s(. hph Thcory.
33 JJlh'sr>l.J )IJ~LVC~IJ~ .~#p-dngy
David M. Goldschmidt
Algebraic Functions
and Projective Curves
Springer
1);lvid hl. Goltlschmidr
ll)A Cenlcr fnr Communications Rcscarch-Princeton
I'rinccton. N.1 08540-3690, llSA
gold((l'tI:lccr.org
Edi~orinlD ood:
S. Axlcr EW. Cihring K.A. Rikt
Mathrrnaticz Department Mathcrn:~ticsk partment Marhcrnatics 1)ep:trtmcnt
S:m I-r;~nciccoS I:I~C Inst th11 Ilniversrty of C:~lrlorni;r,
Univrrsity IJnivcrsity of Michigan 13crkclcy
San I'r:rncisco. CA 94 1.12 Ann Arhor. MI 48 I00 Rcrkelcy. CA q4720-3840
I'SA 15A IISA
axlrrQdwrdu IpchrinpBmath 1~a.umich.cdu [email protected]~~
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1111-y: Ire- wli~vcr1 '1 pr~~priet:~rtr}y!I lls,
To Cherie, Laura, Katie, and Jessica
Preface
This book grew out of a set of notes for a series of lectures I orginally gave at
the Center for Communications Research and then at Princeton University. The
motivationwastotrytounderstandthebasicfactsaboutalgebraiccurveswithout
the modern prerequisite machinery of algebraic geometry. Of course, one might
wellaskifthisisagoodthingtodo.Thereisnoclearanswertothisquestion.In
short,wearetradingoffeasieraccesstothefactsagainstalossofgeneralityand
an impaired understanding of some fundamental ideas. Whether or not this is a
usefultradeoffissomethingyouwillhavetodecideforyourself.
Oneofmyobjectiveswastomaketheexpositionasself-containedaspossible.
Giventhechoicebetweenareferenceandaproof,Iusuallychosethelatter.Al-
though I worked out many of these arguments myself, I think I can confidently
predict that few, if any, of them are novel. I also made an effort to cover some
topics that seem to have been somewhat neglected in the expository literature.
Among these are Tate’s theory of residues, higher derivatives and Weierstrass
pointsincharacteristic p,andinseparableresiduefieldextensions.Forthetreat-
mentofWeierstrasspoints,aswellasakeyargumentintheproofoftheRiemann
Hypothesis for finite fields, I followed the fundamental paper by Sto¨hr–Voloch
[19].Inadditiontothisimportantsource,Ioftenreliedontheexcellentbookby
Stichtenoth[17].
It is a pleasure to acknowledge the excellent mathematical environment pro-
videdbytheCenterforCommunicationsResearchinwhichthisbookwaswritten.
Inparticular,IwouldliketothankmycolleaguesToniBluher,BradBrock,Ev-
erettHowe,BruceJordan,AllanKeeton,DavidLieberman,VictorMiller,David
Zelinsky, and Mike Zieve for lots of encouragement, many helpful discussions,
andmanyusefulpointerstotheliterature.
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Background 1
1.1 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 FunctionFields 40
2.1 DivisorsandAdeles . . . . . . . . . . . . . . . . . . . . . . 40
2.2 WeilDifferentials . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 EllipticFunctions . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 GeometricFunctionFields . . . . . . . . . . . . . . . . . . 54
2.5 ResiduesandDuality . . . . . . . . . . . . . . . . . . . . . 58
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 FiniteExtensions 68
3.1 NormandConorm . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 ScalarExtensions . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 TheDifferent. . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4 SingularPrimeDivisors . . . . . . . . . . . . . . . . . . . . 82
3.5 GaloisExtensions . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 HyperellipticFunctions . . . . . . . . . . . . . . . . . . . . 93
x Contents
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 ProjectiveCurves 103
4.1 ProjectiveVarieties . . . . . . . . . . . . . . . . . . . . . . 103
4.2 MapstoPn . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 ProjectiveEmbeddings . . . . . . . . . . . . . . . . . . . . 114
4.4 WeierstrassPoints . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 PlaneCurves . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5 ZetaFunctions 150
5.1 TheEulerProduct . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 TheFunctionalEquation. . . . . . . . . . . . . . . . . . . . 154
5.3 TheRiemannHypothesis . . . . . . . . . . . . . . . . . . . 156
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A ElementaryFieldTheory 164
References 175
Index 177
Introduction
What Is a Projective Curve?
Classically, a projective curve is just the set of all solutions to an irreducible
( , , )=
homogeneous polynomial equation f X X X 0 in three variables over the
0 1 2
complexnumbers,modulotheequivalencerelationgivenbyscalarmultiplication.
Itisverysafetosay,however,thatthisanswerisdeceptivelysimple,andinfact
liesatthetipofanenormousmathematicaliceberg.
Thesizeoftheicebergisduetothefactthatthesubjectliesattheintersection
ofthreemajorfieldsofmathematics:algebra,analysis,andgeometry.Theorigins
of the theory of curves lie in the nineteenth century work on complex function
theory by Riemann, Abel, and Jacobi. Indeed, in some sense the theory of pro-
jective curves over the complex numbers is equivalent to the theory of compact
Riemannsurfaces,andonecouldlearnafairamountaboutRiemannsurfacesby
specializing results in this book, which are by and large valid over an arbitrary
=C
ground field k, to the case k . To do so, however, would be a big mistake
fortworeasons.First,someofourresults,whichareobtainedwithconsiderable
difficultyoverageneralfield,aremuchmoretransparentandintuitiveinthecom-
plexcase.Second,thetopologicalstructureofcomplexcurvesandtheirbeautiful
relationship to complex function theory are very important parts of the subject
thatdonotseemtogeneralizetoarbitrarygroundfields.Thecomplexcaseinfact
deservesabookalltoitself,andindeedtherearemanysuch,e.g.[15].
Thegeneralizationtoarbitrarygoundfieldsisatwentiethcenturydevelopment,
pioneeredbytheGermanschoolofHasse,Schmidt,andDeuringinthe1920sand
1930s. A significant impetus for this work was provided by the development of
xii Introduction
algebraicnumbertheoryintheearlypartofthecentury,foritturnsoutthatthere
is a very close analogy between algebraic function fields and algebraic number
fields.
TheresultsoftheGermanschoolsetthestageforthedevelopmentofalgebraic
geometry over arbitrary fields, but were in large part limited to the special case
of curves. Even in that case, there were serious difficulties. For example, Hasse
wasabletoprovetheRiemannhypothesisonlyforellipticcurves.Theprooffor
curvesofhighergenuscamefromWeilandmotivatedhisbreakthroughworkon
abstractvarieties.Thisinturnledtothe“greatleapforward”bytheFrenchschool
ofSerre,Grothendiek,Deligne,andotherstothetheoryofschemesinthe1950s
and1960s.
Thefloweringofalgebraicgeometryinthesecondhalfofthecenturyhas,toa
largeextent,subsumedthetheoryofalgebraiccurves.Thisdevelopmenthasbeen
somethingofatwo-edgedsword,however.Ontheonehand,manyoftheresults
oncurvescanbeseenasspecialcasesofmoregeneralfactsaboutschemes.This
providestheusualbenefitsofaunifiedandinsomecasesasimplifiedtreatment,
together with some further insight into what is going on. In addition, there are
some important facts about curves that, at least with the present state of knowl-
edge,canonlybeunderstoodwiththemorepowerfultoolsofalgebraicgeometry.
Forexample,thereareimportantpropertiesoftheJacobianofacurvethatarise
fromitsstructureasanalgebraicgroup.
Ontheotherhand,thefull-blowntreatmentrequiresthestudenttofirstmaster
theconsiderablemachineryofsheaves,schemes,andcohomology,withtheresult
that the subject becomes less accessible to the nonspecialist. Indeed, the older
algebraic development of Hasse et al. has seen something of a revival in recent
years,dueinparttotheemergenceofsomeapplicationsinotherfieldsofmath-
ematics such as cryptology and coding theory. This approach, which is the one
followedinthisbook,treatsthefunctionfieldofthecurveasthebasicobjectof
study.
In fact, one can go a long way by restricting attention entirely to the func-
tion field (see, e.g., [17]), because the theory of function fields turns out to be
equivalenttothetheoryofnonsingularprojectivecurves.However,thisisrather
restrictivebecausemanyimportantexamplesofprojectivecurveshavesingular-
ities.Afeatureofthisbookisthatwegobeyondthenonsingularcaseandstudy
projective curves in general, in effect viewing them as images of nonsingular
curves.
What Is an Algebraic Function?
Forourpurposes,analgebraicfunctionfieldKisafieldthathastranscendencede-
greeoneoversomebasefieldk,andisalsofinitelygeneratedoverk.Equivalently,
( ) ∈
Kisafiniteextensionofk x forsometranscendentalelementx K.Examplesof
suchfieldsabound.Theycanbeconstructedviaelementaryfieldtheorybysim-
