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 PDF
Preview 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
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.
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