Topics in theory of algebraic functional fields

653 Pages·2006·3.551 MB·English

by  Gabriel Daniel Villa Salvador

Mathematics: Theory & Applications
Series Editor
Nolan Wallach
Gabriel Daniel Villa Salvador
Topics in the Theory of
Algebraic Function Fields
Birkha¨user
Boston • Basel • Berlin
GabrielDanielVillaSalvador
CentrodeInvestigacio´nydeEstudiosAvanzadosdelI.P.N.
DepartamentodeControlAutoma´tico
Col.Zacatenco,C.P.07340
Me´xico,D.F.
Me´xico
MathematicsSubjectClassification(2000):11R58,11R60,14H05,11G09,11R32,12F05,12F10,12F15,
11S20,14H55,11R37,11R29,14G10,14G15,14G50,11S31,11S20,14H25,12G05
LibraryofCongressControlNumber:2006927769
ISBN-100-8176-4480-6 e-IBSN0-8176-4515-2
ISBN-13978-0-8176-4480-2
Printedonacid-freepaper.
(cid:1)c2006Birkha¨userBoston
BasedontheoriginalSpanishedition,Introduccio´nalaTeor´ıadelasFunctionesAlgebraicas,Fondode
CulturaEcono´mica,Me´xico,2003
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermis-
sionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233SpringStreet,
NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use
inconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto
proprietaryrights.
PrintedintheUnitedStatesofAmerica. (TXQ/MP)
987654321
www.birkhauser.com
ToMartha,Sof´ıa,andmyfather
Giveamanafishandyoufeedhimforaday.Teachhimhowtofish
andyoufeedhimforalifetime.
LaoTse
Hewhoiscontinuallythinkingthingseasyissuretofindthemdifficult.
LaoTse
Laeducacio´nesunseguroparalavidayunpasaporteparala
eternidad.
(Educationisaninsuranceforlifeandapassportforeternity.)
AparisiyGuijarro
Preface
What are function fields, and what are they useful for? Let us consider a compact
Riemann surface, that is, a surface in which every point has a neighborhood that is
isomorphic to an open set in the complex field C. Now assume the surface under
consideration to be the Riemann sphere S2; then the meromorphic functions defined
in S2,bywhichwemeanfunctionsfrom S2 toC∪{∞}whoseonlysingularitiesare
poles,arepreciselytherationalfunctions f(z),where f(z)andg(z)arepolynomials
g(z)
with coefficients in C. These functions form a field C(z) called the field of rational
functions in one variable over C. In general, if R is a compact Riemann surface, let
usconsiderthemeromorphicfunctionsdefinedon R.Thesetofsuchfunctionsforms
afield,whichiscalledthefieldofmeromorphicfunctionsof R;itturnsoutthatthis
fieldisafiniteextensionofC(z),or,inotherwords,afieldofalgebraicfunctionsof
onevariableoverC.
Now,twoRiemannsurfacesareisomorphicasRiemannsurfacesifandonlyiftheir
respectivefieldsofmeromorphicfunctionsareC-isomorphicfields.Thistellsusthat
such Riemann surfaces are completely characterized by their fields of meromorphic
functions.
Inalgebraicgeometry,letusconsideranarbitraryfieldk,andletC beanonsingu-
larprojectivecurvedefinedonk.ItturnsoutthatthesetofregularfunctionsoverC
isafiniteextensionofthefieldk(x)ofrationalfunctionsoverk.Thisfieldofregular
functionsonC isafieldofalgebraicfunctionsofonevariableoverk.
Thecorrespondencebetweencurvesandfunctionfieldsisasfollows.Assumekto
bealgebraicallyclosed.IfC isanonsingularprojectivecurve,considerthefieldk(C)
consisting of all regular functions in C. Conversely, for a given function field K/k
(seeChapter1),thereexistsanonsingularprojectivecurveC (whichisuniqueupto
isomorphism),suchthatk(C)isk-isomorphictoK.Ontheotherhand,theplaces(see
Chapter2)areinone-to-onecorrespondencewiththepointsofC:toeachpoint P of
C weassociatethemaximalidealm ofthevaluationringϑ .
P P
Thereexistsathirdareaofstudyinwhichfunctionfieldsshowup.Thisisnumber
theory. Here a field of functions of one variable will play a role similar to that of a
viii Preface
finite extension of the field Q of rational numbers. This is the point of view that we
willbeadoptinginthecourseofthisbook.
Thereaderwhoisfamiliarwithelementarynumbertheorymayconsiderthatthe
fieldk(x)ofrationalfunctionsoverk istheanalogueoftherationalfieldQ,thepoly-
nomial ring k[x] is the analogue of the ring of rational integers Z, and finally that a
fieldoffunctionsofonevariableistheanalogueofafiniteextensionofQ.Itturnsout
thattheanalogyismuchstrongerwhenthefieldk isfinite.
Thementionedanalogyworksinbothdirections.Oftentimesaproblemthatgets
posedinnumberfieldsor,inotherwords,infiniteextensionsofQ,admitsananalo-
gousprobleminfunctionfields,andtheotherwayaround.Forexample,ifweconsider
theclassicalRiemannzetafunctionζ(s),itisstillunknownwhetherRiemann’scon-
jecture on nontrivial zeros of ζ(s) holds (although a proof of its validity has been
announced,thishasnotbeenconfirmedyet).Theanalogueofthisprobleminfunction
fieldswassolvedbyWeilinthemiddleofthelastcentury(Chapter7).
Inasimilarway,theclassicaltheoremofKronecker–Weberonabelianextensions
of Q has its analogue in function fields. The Kronecker–Weber theorem establishes
thatanyabelianextensionofQiscontainedinacyclotomicextension.Inotherwords,
themaximalabelianextensionofQistheunionofallitscyclotomicextensions.The
analogue to this result is the theory of Carlitz–Hayes, which establishes, first of all,
theanaloguesinfunctionfieldsoftheusualcyclotomicfields.Themerefactofadding
rootsofunity,asintheclassicalcase,doesnotgetusveryfar,sinceitwouldprovide
usonlywithwhatweshallcallextensionsofconstants,whichisfarawayfromgiving
us all abelian extensions of a rational function field k(T), where k is a finite field.
ThetheoryofCarlitz–Hayes(Chapter12)providesuswiththeauthenticanalogueof
cyclotomicfields,whichleadsustotheequivalenttotheKronecker–Webertheoremin
functionfields.Thissametheorymaybegeneralizedbyconsideringnotonlyk(T)but
also finite extensions. The study of this generalization gives as a result the so-called
Drinfeldmodules,orellipticmodules,asDrinfeldcalledthem.Abriefintroductionto
DrinfeldmoduleswillbepresentedinChapter13.
In the other direction we have Iwasawa’stheoryin numberfields. The originsof
thistheoryaresimilar(innumberfields)toconsideringacurveoverafinitefieldand
extendingthefieldofconstantsk toitsalgebraicclosure;inordertodothisonemust
adjoinallrootsofunity.Inthenumberfieldcase,adjoiningallrootsofunitygivesa
fieldtoobig,andforthisreasononemustconsideronlyrootsofunitywhoseorderis
a power of a given prime number. In this way, Iwasawa obtained the Z -cyclotomic
p
extensionsofnumberfields,whereZ istheringof p-adicintegers.
p
In the study of function fields, one may put the emphasis on the algebraic–
arithmetic aspects or on the geometric–analytic ones. As Claude Chevalley rightly
pointsoutinhisbook[22],itisabsolutelynecessarytostudybothaspectsofthethe-
ory,sinceeachonehasitsownstrengthsinanaturalway.However,eventhoughboth
viewpoints may be treated in a textbook, one of them must be selected as the main
focus of the book, since keeping both at the same time would be like superposing
twophotographsofthesameobjecttakenfromdifferentangles;theresultwouldbea
blurredanddullimageoftheobject.
Preface ix
Our point of view in all the book will be the algebraic–arithmetic approach, and
ourprincipalinterestwillbethestudyoffunctionfieldsaspartofthealgebraictheory
of numbers. This by no means should be interpreted in the sense that we consider
unimportanttheanalyticandthegeometricapproaches.
Aswementionedbefore,whenthebasefieldk ofafunctionfieldisafinitefield,
theanalogybetweenthesefieldsandnumberfieldsismuchcloser.Inthissituationit
ispossibletodefinezetafunctions, L-series,classnumbers,etc.However,itmustme
stressed that there are fundamental differences between these two families of fields:
the number fields have archimedean absolute values and the function fields do not
(seeChapter2);theringofrationalintegersZandtherationalfieldQareessentially
unique,asopposedtopolynomialringsk[x]andrationalfunctionfieldsk(x),which
arerespectivelyisomorphictomanyringsandfields.Consequently,thesituationofZ
beingcontainedinQadmitsnotonlyoneanalogueinfunctionfields,butaninfinity
ofthem.Therefore,itisveryimportanttokeepinmindbothaspects:thesimilarities
betweenbothfamiliesoffieldsaswellastheirfundamentaldifferences.
This book may be used for a first-year graduate course on number theory. We
tried to make it self-contained whenever possible, the only prerequisites being the
following: a basic course in field theory; a first course in complex analysis; some
basic knowledge of commutative algebra, say at the level of the Atiyah–Macdonald
book [4]; and the mathematical maturity required to learn new concepts and relate
themtoknownones.
The first four chapters can be used for an introductory undergraduate course for
mathematics majors, and Chapters 5, 6, 7, and 9 for a second course, avoiding the
mosttechnicalparts,forinstancetheproofsoftheRiemannhypothesis,Cˇebotarev’s
densitytheorem,thecomputationofthedifferent,andTate’sgenusformula.
The introductory chapter was written mainly to motivate the study of transcen-
dentalextensions,absolutevaluesofQ,andcompactRiemannsurfaces.However,in
ordertoavoidmakingitlongandtedious,wewillestablishtheresultsneededforeach
topic at the moment they are required. The reason for this selection is as follows. A
function field K over k is really just a finitely generated transcendental extension of
k,withtranscendencedegreeone.Ontheotherhand,thestudyofsuchfieldsleadsus
to the study of their absolute values, whose analogues are, up to a certain point, the
absolute values in Q. Finally, compact Riemann surfaces constitute a splendid geo-
metricrepresentationoffunctionfields.InthecaseofRiemannsurfacesweshallnot
provideproofsofthepresentedresults,sinceourinterestisonlythatthereaderknow
thefundamentalresultsoncompactRiemannsurfaces,andusethemasamotivation
tostudymoregeneralsituations.
Chapter2istheintroductiontoourmainobjective.There,wedefinegeneralcon-
ceptsthatwillbenecessaryinthecourseofthisvolume,suchasfieldsofconstants,
valuations,places,valuationrings,absolutevalues,etc.Oncetheseconceptsaremas-
tered,weshallstudythecompletionsofafieldwithrespecttoanabsolutevalue.The
usefulness of the study of completions with respect to a metric is well known in the
area of analysis. In our case, we shall use these completions as a basic tool for the
study of the arithmetic properties of places in field extensions (Chapter 5). For this
chapter it is convenient, but not necessary, that the reader be familiar with the com-
x Preface
pletionofametricspaceoratleastwiththestandardcompletionofQwithrespectto
theusualabsolutevalueobtainingthefieldofrealnumbersR.Wefinishthechapter
with Artin’s approximation theorem, which can be considered as the generalization
oftheChineseremaindertheoremandwhichestablishesthefollowing:Givenafinite
numberofabsolutevaluesandanequalnumberofelementsofthefield,wecanfind
an element of the field that approximates the given elements in each absolute value
as much as we want. Theorem 2.5.20 is the characterization of the completion of a
functionfield.
Chapter3isdedicatedtothefamousRiemann–Rochtheorem(Theorem3.5.4and
corollaries) which is, without any doubt, the most important result of our book. The
Riemann–RochTheoremstatestheequalitybetweendimensionsofvectorspaces,de-
greeofafieldextensionandaveryimportantfieldinvariant:thegenus.Inordertoes-
tablishtheRiemann-RochTheoremonerequiresvariouspreliminaryconcepts,which
willbedefinedinthischapterandwillplayacentralroleintherestofthebook:divi-
sors,adelesorrepartitions,Weildifferentials,classgroups,etc.Thewholetheoryof
functionfieldsdependsheavilyontheRiemann–Rochtheorem.
An important part of the work of any mathematician at any level is to develop
and know examples concerning the topic on which he or she is working. Chapter 4
is dedicated to giving examples of the results found in Chapter 2 and 3. In the first
twosectionswepresentexamplesandcharacterizethefunctionfieldsofgenus0and
1respectively,andinthelastsectionwecalculatethegenusofaquadraticextension
of a rational function field. Even though the genus can be found much more easily
using the Riemann–Hurwitz genus formula (Theorem 9.4.2), the methods we use in
thischapterarevaluablebythemselves.
Chapter5dealswithGaloistheoryoffunctionfields.AfterChapter3,thischapter
can be considered as the second in importance. It is dedicated to the arithmetic of
functionfields(decompositionofplacesintheextensions,ramification,inertia,etc.).
Herewestudytherelationshipbetweenthedecompositionofplacesinanextensionof
function fields and the decomposition in the corresponding completions. Section 5.6
containsmanytechnicaldetailsnecessarytounderstandthenotionofadifferentinan
extensionandthedifferentinanextensionofDedekinddomains,whichisthewaywe
studythearithmeticofnumberfields(Theorem5.7.12).Thelastsectionofthechapter
concernsthestudyofthedifferentbymeansofthelocaldifferents(Theorem5.7.21).
Theproofcanbeomittedwithoutanylossofcontinuity.Weendthischapterwithan
introductiontoramificationgroups.
Chapter 6 deals with congruence function fields, that is, function fields whose
constantfieldisfinite.Aswesaidpreviously,theanalogybetweenthiskindoffunction
fieldsandnumberfieldsismuchcloser.InthischapterwestudyzetafunctionsandL-
series,aswellastheirfunctionalequations.
Chapter 7 is dedicated to the Riemann hypothesis in function fields (Theorem
7.2.9). The proof that we present here is essentially due to Bombieri [7]. The reader
canomitthedetailsoftheproofwithoutanylossofcontinuity.Asanapplicationof
theRiemannhypothesiswepresentanestimationonthenumberofprimedivisorsin
a congruence function field, as well as the determination of the fields of class num-
ber1.
Preface xi
Chapter 8 studies constant extensions in general, a particular case of which was
seeninChapter6,namelythecasethattheconstantfieldisfinite.Wehavepreferredto
presentfirstthisspecialcaseforthereadersthatareinterestedinthemostusualcases,
that is, when the constant field is a perfect field, in order to avoid all the technical
detailsofthegeneralcase.Inthischapterwestudytheconceptsofseparabilityandof
aseparablygeneratedfieldextension.Wealsostudythegenuschangeinthiskindof
extensionandwillseethatthegenusofthefielddecreases.
Chapter9concernstheRiemann–Hurwitzgenusformulaforgeometricandsepa-
rableextensions,whichisprobablythebesttechniqueforcalculatingthegenusofan
arbitraryfunctionfield.Forinseparableextensions,Tate[152]usedasubstituteforthe
ordinary trace and found a genus formula for this type of extension. That substitute
is the one used in the Riemann–Hurwitz formula. In Section 9.5, we present Tate’s
results. In the last section of the chapter, we revisit function fields of genus 0 and
1 and present the automorphism group of elliptic function fields. We conclude with
hyperellipticfunctionfields,whichwillbeusedinChapter10forcryptosystems.
InChapter10weapplythetheoryoffunctionfields,especiallyChapter6and7,to
cryptography.Webeginwithabriefgeneralintroductiontocryptography:symmetric
and asymmetric systems, public-key cryptosystems, the discrete logarithm problem,
etc.Oncetheseconceptsareintroducedweapplythetheoryofellipticandhyperellip-
ticfunctionfieldstocryptosystems.Inthisway,weshallseethatsomegroupsthatare
determinedbyellipticfunctionfields,aswellassomeJacobians,maybeusedbothfor
public-keycryptosystemsandfordigitalsignaturesandauthentication.
Chapter11isabriefintroductiontoclassfieldtheory.WestudyCˇebotarev’sden-
sitytheoremandbrieflyintroduceprofinitegroups.Finallywepresent,withoutproofs,
basicresultsofglobalaswellaslocalclassfieldtheory.Theseresultswillbeusedin
Chapter 12 to prove Hayes’s theorem, which is analogous to the Kronecker–Weber
theorem on the maximal abelian extension of a congruent function field, that is, a
functionfieldwhoseconstantfieldisfinite.
Chapter12isdedicatedtothetheoryofcyclotomicfunctionfieldsduetoL.Carlitz
and D. Hayes [15, 61]. We shall see that these fields are the analogue of the usual
cyclotomicfields.
In Chapter 13 we give a brief introduction to Drinfeld, or elliptic, modules. The
original objective of Drinfeld’s module theory was to generalize the analogue of the
Kronecker–Weber theorem to a function field over a general finite field, as well as
complexmultiplicationandellipticcurves.WebeginbypresentingtheCarlitzmodule,
whichisstudiedinChapter12andisthesimplestDrinfeldmodule.Usingtheanalytic
theory of exponential functions and lattices, we shall see that Drinfeld modules are
ubiquitous.Ontheotherhand,thesemodulesprovideuswithanexplicitclasstheory
forgeneralfunctionfieldsoverafinitefield.Weendthechapterwiththeapplication
ofDrinfeldmodulestocryptography.
The last chapter is a study of the automorphism group of a function field. First
wegiveanotionofdifferentiationduetoH.HasseandF.Schmidt[58]andthenwe
useittostudytheWronskiandeterminantandWeierstrasspointsincharacteristic p.
Wewillseethatthebehaviorincharacteristic pisdifferentfromthatincharacteristic
0. We will use Weierstrass points to prove the classical result about the finiteness of