; TeX output 2002.11.14:1829 y html: html:i html: html: cJN q cmbx12Computations
ondiagonalquartico~ surfacescjiBN G cmbx12MartinzJamesBrighut iClarezCollege !нXQ ff cmr12A/dissertationsubmittedforthedegreeofDodCctorof>DPhilosophy/attheUniversityofCambridgeZC
SeptembdCer/2002 * y html: html:i @!$ HPreface XQ cmr12ThisvdissertationisnotsubstanrtiallythesameasanyIvhavesubmittedfor adegreeordiplomaorotherqualicationatanryotheruniversityV.Except whereindicatedinthetext,Litistheresultofmryownworkandcontains nothingwhicrhistheoutcomeofworkdoneincollabSoration.0IFIthankFsmryresearchsupServisor,gJSirPeterSwinnerton-Dyer,gJforhisexpSert guidancePinthecourseofmrystudies.mInparticular,hehasmademerealisethe greatvXalueofexplicitcalculationsinunderstandingnrumbSertheoryV.(Iwould also'GlikretothankthemanypSeoplewhohavehelpSedmewithmathematical discussions,qaswrellasthosewhohavegivenmoralsuppSort.ڞIϛacknowledgethe nancial@suppSortoftheEngineeringandPhrysicalSciencesResearchCouncil. I Sam aalsovrerygratefultoIBMResearcrhforawardingmetheirrstSpSecial EuropSeanJResearcrhFVellowship,jincludinganIBMJThinkPadcomputerwhich hasbSeeninrvXaluablebothforcomputationandforproducingthisdissertation. FinallyV,ZIwrouldliketothanktheauthorsofPVARI/GPformakingthefruits oftheirwrorkfreelyavXailable.gCC"ClareCollegeC"SeptemrbSer2002 , y html: html:xiii @!$
?Summary0WVeareconcernedwiththesolubilitryinintegersoftheequation 퍒 g cmmi12a|{Y cmr80X 4ڍ0\/+a̿1X 4ڍ1+a̿2X 4ڍ2+a̿3X 4ڍ3 ٹ=UR0 where&a̿0,ta̿1,a̿2 and&a̿3arexedinrtegralcoSecients.%Astheequationis homogeneous,Jrthis"disequivXalenrttostudyingrationalsolutions.Thisequation denesEasurfaceinthree-dimensionalpro jectivrespace"
msbm10P23,[whichwedenote bryVp.8GeometricallyV,VisaK3surface.0OneconditionwhicrhisclearlynecessaryforVatohaverationalpSointsis thatiGtherebSepoinrtsonVineachcompletionQ2 cmmi8v VTofQ.VVarietiesforwhich thisconditionissucienrtaresaidtosatisfythe%@ cmti12Hassefprinciple.(CIngeneral, theHasseprincipledoSesnothold,
obutmanrycounterexamplesareexplained brytheso-calledBrffauer{ManinRobstruction.