Integral points on hyperelliptic curves

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1、Algebra&NumberTheoryVolume22008No.8IntegralpointsonhyperellipticcurvesYannBugeaud,MauriceMignotte,SamirSiksek,MichaelStollandSzabolcsTengelymathematicalsciencespublishersALGEBRAANDNUMBERTHEORY2:8(2008)IntegralpointsonhyperellipticcurvesYannBugeaud,MauriceMignotte,SamirSiksek,MichaelSt

2、ollandSzabolcsTengelyLetCVY2DaXnC


Cabeahyperellipticcurvewiththearationalinte-n0igers,n5,andthepolynomialontheright-handsideirreducible.LetJbeitsJacobian.WegiveacompletelyexplicitupperboundfortheintegralpointsonthemodelC,providedweknowatleastonerationalpointonCandaMordell–Weilbasis

3、forJ.Q/.WealsoexplainapowerfulrefinementoftheMordell–Weilsievewhich,combinedwiththeupperbound,iscapableofdeterminingalltheintegralpoints.Ourmethodisillustratedbydeterminingtheintegralpointson

thegenus2hyperellipticmodelsY2YDX5XandYDX.251.IntroductionConsiderthehyperellipticcurvewi

4、thaffinemodelCVY2DaXnCaXn1C


Ca;(1-1)nn10witha0;:::;anrationalintegers,an6D0,n5,andthepolynomialontheright-handsideirreducible.LetHDmaxfja0j;:::;janjg.Inoneoftheearliestappli-cationsofhistheoryoflowerboundsforlinearformsinlogarithms,Baker[1969]showedthatanyintegralpoint.X;Y/onthisa

5、ffinemodelsatisfies2
max.jXj;jYj/expexpexp.n10nH/n:Suchboundshavebeenimprovedconsiderablybymanyauthors,includingSprin-dzuk[1977],Brindza[1984],Schmidt[1992],Poulakis[1991],Bilu[1995],Bu-ˇgeaud[1997]andVoutier[1995].Despitetheimprovements,theboundsremainastronomicalandofteninvolveinexp

6、licitconstants.Inthispaperweexplainanewmethodforexplicitlycomputingtheintegralpointsonaffinemodelsofhyperellipticcurves(1-1).Themethodfallsintotwodistinctsteps:MSC2000:primary11G30;secondary11J86.Keywords:curve,integralpoint,Jacobian,height,Mordell–Weilgroup,Baker’sbound,Mordell–Weilsi

7、eve.859860Y.Bugeaud,M.Mignotte,S.Siksek,M.StollandSz.Tengely(i)Wegiveacompletelyexplicitupperboundforthesizeofintegralsolutionsof(1-1).ThisupperboundcombinesmanyrefinementsfoundinthepapersofVoutier,Bugeaud,andothers,togetherwithMatveev’sbounds[2000]forlinearformsinlogarithms,andamethod

8、forbo