On Some Applications of Diophantine Approximations: a translation of Carl Ludwig Siegel’s Über einige Anwendungen diophantischer Approximationen by Clemens Fuchs, with a commentary and the article Integral points on curves: Siegel’s theorem after Siegel’s

169 Pages·2014·6.69 MB·English

by  Umberto Zannier (eds.)

On Some Applications
of Diophantine
Approximations
(a translation of Carl Ludwig Siegel’s U¨ ber einige
AnwendungendiophantischerApproximationen
by Clemens Fuchs)
edited by
Umberto Zannier
with a commentary and the article Integral
points on curves: Siegel’s theorem after Siegel’s
proof
EDIZIONI
DELLA
NORMALE
2
QUADERNI
MONOGRAPHS
UmbertoZannier
ScuolaNormaleSuperiore
PiazzadeiCavalieri,7
56126Pisa,Italia
ClemensFuchs
UniversityofSalzburg
DepartmentofMathematics
Hellbrunnerstr.34/I
5020Salzburg,Austria
OnSomeApplicationsofDiophantineApproximations
On Some Applications
of Diophantine
Approximations
(a translation of Carl Ludwig Siegel’s U¨ber einige
AnwendungendiophantischerApproximationen
by(cid:0)Clemens Fuchs)
edited by
Umberto Zannier
with a commentary and the article Integral
points on curves: Siegel’s theorem after Siegel’s
proof by Clemens Fuchs and Umberto Zannier
(cid:2)c 2014ScuolaNormaleSuperiorePisa
ISBN978-88-7642-519-6
ISBN978-88-7642-520-2 (eBook)
Contents
Preface vii
C(cid:76)(cid:69)(cid:77)(cid:69)(cid:78)(cid:83)(cid:0)(cid:38)(cid:85)(cid:67)(cid:72)(cid:83)
OnsomeapplicationsofDiophantineapproximations 1
1 PartI:Ontranscendentalnumbers 8
1 Toolsfromcomplexanalysis . . . . . . . . . . . . . . . . . . . 8
2 Toolsfromarithmetic . . . . . . . . . . . . . . . . . . . . . . . 19
3 Thetranscendenceof J (ξ) . . . . . . . . . . . . . . . . . . . . 27
0
4 Furtherapplicationsofthemethod . . . . . . . . . . . . . . . . 31
2 PartII:OnDiophantineequations 47
1 Equationsofgenus0 . . . . . . . . . . . . . . . . . . . . . . . 52
2 Idealsinfunctionfieldsandnumberfields . . . . . . . . . . . . 54
3 Equationsofgenus1 . . . . . . . . . . . . . . . . . . . . . . . 59
4 AuxiliarymeansfromthetheoryofABELfunctions . . . . . . . 61
5 Equationsofarbitrarypositivegenus . . . . . . . . . . . . . . . 65
6 Anapplicationoftheapproximationmethod . . . . . . . . . . . 68
7 Cubicformswithpositivediscriminant . . . . . . . . . . . . . . 74
C(cid:65)(cid:82)(cid:76)L(cid:14)(cid:0)Siegel
U¨bereinigeAnwendungendiophantischerApproximationen 81
ClemensFuchsandUmberto Zannier
Integralpointsoncurves:
Siegel’stheoremafterSiegel’sproof 139
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2 SomedevelopmentsafterSiegel’sproof . . . . . . . . . . . . . 139
3 Siegel’sTheoremandsomepreliminaries . . . . . . . . . . . . 144
4 ThreeargumentsforSiegel’sTheorem . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Preface
In1929CarlLudwigSiegelpublishedthepaperU¨bereinigeAnwendungendi-
ophantischerApproximationen(appearedinAbh. Preuß. Akad. Wissen. Phys.-
math.Klasse,1929andreproducedinSiegel’scollectedpapersGes.Abh.Bd.I,
Springer-Verlag1966,209-266). ItwasdevotedtoDiophantineApproximation
andapplicationsofit,andbecamealandmarkwork,alsoconcerninganumber
of related subjects. Siegel’s paper was written in German and this volume is
devotedtoatranslationofitintoEnglish(includingalsotheoriginalversionin
German).
Siegel’s paper introduced simultaneously many new methods and ideas. To
commentonthisinsomedetailwouldoccupyafurtherpaper(orseveralpapers),
and here we just limit ourselves to a brief discussion. The paper is, roughly,
dividedintotwoparts:
(a)Thefirstpartwasdevotedtoprovingtranscendenceofnumbersobtained
asvalues(atalgebraicpoints)ofcertainspecialfunctions(includinghypergeo-
metric functions and Bessel functions). In this realm, Siegel’s paper system-
atically developed ideas introduced originally by Hermite in dealing with the
exponential function; a main point is to approximate with rational functions
a function expressed by a power series, and to draw by specialisation numer-
icalapproximationstoitsvalues. Thesenumericalapproximations, ifaccurate
enough,allow,throughstandardcomparisonestimates,toprovetheirrationality
(ortranscendence)ofthevaluesofthefunctioninquestion.
Siegelexploitedandextendedthisprincipleingreatdepth, obtainingresults
whichwere(andare)spectacularlygeneralinthetopic,especiallyatthattime.
Probablythiswasthefirstpapergivingtotranscendencetheorysomecoher-
ence.
Thisstudyalsointroducedrelatedconcepts,liketheoneof‘E-function’and
of‘G-function’(cf.[1,8]),andlednaturallytoalgebraicalandarithmeticalstud-
ies on systems of linear differential equations (with polynomial coefficients),
whicheventuallyinspiredseveraldifferentdirectionsofresearch,allimportant
anddeep.
(b) The second part was devoted to diophantine equations, more precisely
to the study of integral points o algebraic curves. In this realm too, the paper
viii Preface
introduced several further new ideas, also with respect to previous important
papersofSiegel.1 Someinstancesforthenewideascontainedinthesecondpart
are:
– The paper used the embedding of a curve (of positive genus) into its Jac-
obian, and the finite generation of the rational points in this last variety
(whichhadbeenprovedbyL.J.MordellforellipticcurvesandbyA.Weil
ingeneral); thisprovidedabasicinstanceoftheintimateconnectionofdi-
ophantine analysis with algebraic geometry and complex analysis, which
becameunavoidablesincethattime.
– It used and developed the concept of ‘height’ of algebraic points and its
propertiesrelatedtorationaltransformations,especiallyonalgebraiccurves;
this went sometimes beyond results by A. Weil, who had introduced the
conceptandhadalsopointedouttransformationproperties.
Again,thisrepresentedoneoftheveryfirstexamplesofhowthelinkbetween
arithmetic and geometry can be used most efficiently, leading to profound
results.
– Itexploitedtoanewextentthediophantineapproximationtoalgebraicnum-
bers.ThishadbeenusedbyA.Thuearound1909inthecontextofthespecial
curvesdefinedbytheso-called‘Thueequations’. Forgeneralcurves,thedi-
ophantineapproximationdrawnfromintegersolutionsisnotsharpenoughto
providedirectlythesoughtinformation,henceSiegelhadtogomuchdeeper
intothis. Bytakingcoversofthecurve(insideitsJacobian)heimprovedthe
Diophantineapproximation;thenhewasabletoconcludethroughasuitable
refinementofThue’stheorem,alsoprovedinthepaper. Thisrefinementwas
notasstrongasK.F.Roth’stheorem(provedonlyin1955),creatingfurther
complications to the proof; Siegel had to use simultaneous approximations
toseveralnumberstogetasharpenoughestimate.(Forthistask,Siegelused
ideasintroducedinthefirstpartofthepaper.)
On combining all of this, Siegel produced his theorem on integral points on
curves, which may be seen as a final result in this direction. At that time, this
was especially impressive, since Diophantine equations were often treated by
adhocmethods,withlittlepossibilityofembracingwholefamilies. (Oneofthe
fewexceptionsoccurredwithThue’smethods,alludedtoabove.)
The theorem also bears a marked geometrical content: an affine curve may
haveinfinitelymanyintegralpointsonlyifithasnon-negativeEulercharacter-
istic. (Thisisdefinedas2−2g−s wheregisthegenusofasmoothprojective
modelofthecurveandwheresisthenumberofpointsatinfinity,namelythose
1Forinstance,Siegelhadwritten(chronologically)rightbeforeanotherremarkablepaperonin-
tegralpointsbyprovingtheirfinitenessforhyperellipticequations y2 = f(x)underappropriate
assumptions(thisresultwasextractedfromalettertoMordellandwaspublishedunderthepseud-
onymXin[47]).
ix Preface
missing on the affine curve with respect to a smooth complete model.) Con-
versely, this condition becomes sufficient (for the existence of infinitely many
integral points) if we allow a sufficiently large number field and a sufficiently
large(fixed)denominatorforthecoordinates.2
Eachofthenumerousresultsandideasthatwehavementionedwouldhave
representedatthattimeamajoradvanceinitself. Henceitisdifficulttooveres-
timatetheimportanceofthispaperanditsinfluence,eventhinkingofcontem-
porarymathematics.
The paper, however, being written in German, is not accessible for a direct
reading to all mathematicians. There are of course modern good or excellent
expositionsinEnglishofsomeoftheresults, howeverwebelieveitmaybeof
interestformanytogothroughtheoriginalsourceforapreciseunderstanding
ofsomeprinciples,asreallyconceivedbySiegel. Inaddition,wethinkthatthis
paperisamodelalsofrom theviewpointofexposition; theideasandmethods
arepresentedinalimpidandsimultaneouslypreciseway.
AllofthisledthesecondauthortotheideaoftranslatingthepaperintoEng-
lish, and of publishing the result by the ‘Edizioni della Normale’. After some
partialattemptsforatranslation,thiswasfinallycarriedoutbythefirstauthor,
andhereistheoutput. Thetranslationwasmadeliterallyandkeepingthestyle
usedbySiegelasfaraspossible,insteadofrephrasingthetextinmoremodern
style. (Inparticular, Siegel is not usingtheengaging‘we’ but insteaduses the
comprehensive ‘one’. The reader should have this in mind when reading the
translation.)
Ithasbeeneventuallypossiblealsotoincludetheoriginaltextinthisvolume,
whichprovidesadditionalinformation.
Wehavealsoaddedtothetranslatedtextasmallnumberoffootnotes(marked
as“FOOTNOTE BY THE EDITORS”)tohighlightafewpointsthatwethinkare
worthnotingfortheconvenienceofthereader. Further,afterthetranslation,we
haveincludedintothispublicationanarticlebythetwoofus,describingsome
developmentsinthetopicofintegralpoints,andthreemodernproofsofSiegel’s
theorem (two of them being versions of the original argument). We have also
insertedafewreferencestootherworkarisingfromthepaperofSiegel.
ACKNOWLEDGEMENTS. WethanktheEdizionidellaNormaleforhavingwel-
comethisproject. WeespeciallythankMrs. LuisaFerriniofthe‘CentroEdiz-
ioni’, whosegreatcareandattentionhavemadeitpossibletoreachthesought
goal.
ClemensFuchsandUmbertoZannier
2Notethataffineimpliess >0;however,inviewofFaltings’theorem-see2.1below-allof
thisremainstruealsoforprojectivecurves,i.e.whens=0.
On some applications
of Diophantine approximations*
ClemensFuchs
EssaysofthePrussianAcademyofSciences.
Physical-MathematicalClass1929,No.1
Thewell-knownsimpledeductionruleaccordingtowhichforanydistributionof
morethannobjectstondrawersatleastonedrawercontainsatleasttwoobjects,
givesrisetoageneralizationoftheEuclideanalgorithm,whichbyinvestigations
dueto DIRICHLET, HERMITE and MINKOWSKI turnedouttobethesourceof
importantarithmeticlaws. Inparticularitimpliesastatementonhowprecisely
thenumber0canbeatleastapproximatedbyalinearcombination
L =h ω +···+h ω
0 0 r r
of suitable rational integers h ,...,h , which in absolute value are
0 r
boundedbyagivennaturalnumber H anddonotvanishsimultaneously,andof
givennumbersω ,...,ω ;infactforthebestapproximationitcertainlyholds
0 r
|L|≤(|ω |+···+|ω |)H−r,
0 r
anassertionthatdoesnotdependondeeperarithmeticpropertiesofthenumbers
ω ,...,ω .
0 r
The expression L is called an approximation form. If one then asks, how
preciselythenumber0canatbestbeapproximatedbytheapproximationform
h ω +···+h ω ,thenobviouslyanynon-trivialanswerwillcertainlydepend
0 0 r r
onthearithmeticpropertiesofthegivennumbersω ,...,ω .
0 r
Thisquestionparticularlycontainstheproblemtoinvestigatewhetheragiven
number ω is transcendental or not; one just has to choose ω = 1,ω =
0 1
ω,...,ω = ωr,H = 1,2,3,...,r = 1,2,3,.... The additional assump-
r
tion, to come up even with a non-zero lower bound for |L| as a function in H
andr,givesapositiveturnaroundtothetranscendenceproblem.
Also,theupperboundforthenumberoflatticepointsonanalgebraiccurve,
thus in particular the study of the finiteness of this number, leads, as will turn
out later, to the determination of a positive lower bound for the absolute value
ofacertainapproximationform.
Analogoustothearithmeticproblemsofbounding|L|fromaboveandfrom
below is an algebraic question. Let ω (x),...,ω (x) be series in powers of a
0 r
∗CARL LUDWIG SIEGEL, U¨ber einige Anwendungen diophantischer Approximationen, In:
“GesammelteAbhandlungen”,BandI,Springer-Verlag,Berlin-Heidelberg-NewYork,1966,209–
266.