Elliptic Curves

510 Pages·1987·3.438 MB·English

by  Dale Husemöller (auth.)

Elliptic Curves,
Second Edition
Dale Husemöller
Springer
111
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
Springer
New York
Berlin
Heidelberg
Hong Kong
London
Milan
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Dale Husemöller
Elliptic Curves
Second Edition
With Appendices by Otto Forster, Ruth Lawrence, and
Stefan Theisen
With 42 Illustrations
Dale Husemöller
Max-Planck-Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn
Germany
[email protected]
Editorial Board:
S. Axler F.W. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hall University of California,
University University of Michigan Berkeley
San Francisco, CA94132 Ann Arbor, MI 48109 Berkeley, CA94720-3840
USA USA USA
[email protected] [email protected] [email protected]
Mathematics Subject Classification (2000): 14-01, 14H52
Library of Congress Cataloging-in-Publication Data
Husemöller, Dale.
Elliptic curves.— 2nd ed. / Dale Husemöller ; with appendices by Stefan Theisen, Otto Forster, and
Ruth Lawrence.
p. cm. — (Graduate texts in mathematics; 111)
Includes bibliographical references and index.
ISBN 0-387-95490-2 (alk. paper)
1.Curves, Elliptic. 2.Curves, Algebraic. 3.Group schemes (Mathematics) I.Title. II.
Series.
QA567 .H897 2002
516.3′52—dc21 2002067016
ISBN 0-387-95490-2 Printed on acid-free paper.
©2004, 1987 Springer-Verlag New York, Inc.
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To
Robert
and the memory of
Roger,
with whom I first learned
the meaning of collaboration
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Preface to the Second Edition
Thesecondeditionbuildsonthefirstinseveralways.Therearethreenewchapters
whichsurveyrecentdirectionsandextensionsofthetheory,andtherearetwonew
appendices. Then there are numerous additions to the original text. For example, a
very elementary addition is another parametrization which the author learned from
DonZagier y2 = x3−3αx +2β ofthebasiccubicequation.Thisparametrization
isusefulforadetaileddescriptionofellipticcurvesovertherealnumbers.
ThethreenewchaptersareChapters18,19,and20.Chapter18,onFermat’sLast
Theorem, is designed to point out which material in the earlier chapters is relevant
as background for reading Wiles’ paper on the subject together with further devel-
opments by Taylor and Diamond. The statement which we call the modular curve
conjecturehasalonghistoryassociatedwithShimura,Taniyama,andWeiloverthe
last fifty years. Its relation to Fermat, starting with the clever observation of Frey
endinginthecompleteproofbyRibetwithmanycontributionsofSerre,wasalready
mentionedinthefirstedition.TheproofforabroadclassofcurvesbyWileswassuf-
ficienttoestablishFermat’slasttheorem.Chapter18isanintroductiontothepapers
onthemodularcurveconjectureandsomeindicationoftheproof.
Chapter19isanintroductiontoK3surfacesandthehigherdimensionalCalabi–
Yau manifolds. One of the motivations for producing the second edition was the
utilityofthefirsteditionforpeopleconsideringexamplesoffibringsofthreedimen-
sional Calabi–Yau varieties. Abelian varieties form one class of generalizations of
ellipticcurvestohigherdimensions,andK3surfacesandgeneralCalabi–Yaumani-
foldsconstituteasecondclass.
Chapter20isanextensionofearliermaterialonfamiliesofellipticcurveswhere
the family itself is considered as a higher dimensional variety fibered by elliptic
curves.Thefirsttwocasesareonedimensionalparameterspaceswherethefamilyis
twodimensional,hence a surface two dimensional surface parameterspaceswhere
thefamilyisthreedimensional.Thereisthequestionof,givenasurfaceorathree
dimensionalvariety,doesitadmitafibrationbyellipticcurveswithafinitenumber
of exceptional singular fibres. This question can be taken as the point of departure
fortheEnriquesclassificationofsurfaces.
viii PrefacetotheSecondEdition
There are three new appendices, one by Stefan Theisen on the role of Calabi–
YaumanifoldsinstringtheoryandonebyOttoForsterontheuseofellipticcurves
incomputingtheoryandcodingtheory.Inthethirdappendixwediscusstheroleof
ellipticcurvesinhomotopytheory.Inthesethreeintroductionsthereadercangeta
cluetothefar-reachingimplicationsofthetheoryofellipticcurvesinmathematical
sciences.
During the final production of this edition, the ICM 2002 manuscript of Mike
Hopkins became available. This report outlines the role of elliptic curves in homo-
topy theory. Elliptic curves appear in the form of the Weierstasse equation and its
relatedchangesofvariable.Theequationsandthechangesofvariablearecodedin
analgebraicstructurecalledaHopfalgebroid,andthisHopfalgebroidisrelatedto
acohomologytheorycalledtopologicalmodularforms.Hopkinsandhiscoworkers
have used this theory in several directions, one being the explanation of elements
instablehomotopyuptodegree60.Inthethirdappendixweexplainhowwhatwe
described in Chapter 3 leads to the Weierstrass Hopf algebroid making a link with
Hopkins’paper.
Max-Planck-Institutfu¨rMathematik DaleHusemo¨ller
Bonn,Germany
Preface to the First Edition
Thebookdividesnaturallyintoseveralpartsaccordingtothelevelofthematerial,
thebackgroundrequiredofthereader,andthestyleofpresentationwithrespectto
detailsofproofs.Forexample,thefirstpart,toChapter6,isundergraduateinlevel,
thesecondpartrequiresabackgroundinGaloistheoryandthethirdsomecomplex
analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A
general outline of much of the material can be found in Tate’s colloquium lectures
reproducedasanarticleinInventiones[1974].
ThefirstpartgrewoutofTate’s1961HaverfordPhilipsLecturesasanattemptto
writesomethingforpublicationcloselyrelatedtotheoriginalTatenoteswhichwere
moreorlesstakenfromthetaperecordingofthelecturesthemselves.Thisincludes
parts of the Introduction and the first six chapters. The aim of this part is to prove,
byelementarymethods,theMordelltheoremonthefinitegenerationoftherational
pointsonellipticcurvesdefinedovertherationalnumbers.
In 1970 Tate returned to Haverford to give again, in revised form, the original
lecturesof1961andtoextendthematerialsothatitwouldbesuitableforpublication.
Thisledtoabroaderplanforthebook.
The second part, consisting of Chapters 7 and 8, recasts the arguments used in
theproofoftheMordelltheoremintothecontextofGaloiscohomologyanddescent
theory.ThebackgroundmaterialinGaloistheorythatisrequiredissurveyedatthe
beginnngofChapter7fortheconvenienceofthereader.
The third part, consisting of Chapters 9, 10, and 11, is on analytic theory. A
backgroundincomplexanalysisisassumedandinChapter10elementaryresultson
p-adic fields, some of which were introduced in Chapter 5, are used in our discus-
sionofTate’stheoryof p-adicthetafunctions.ThissectionisbasedonTate’s1972
HaverfordPhilipsLectures.
Max-Planck-Institutfu¨rMathematik DaleHusemo¨ller
Bonn,Germany