arithmetic elliptic curves in general position

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1、ARITHMETICELLIPTICCURVESINGENERALPOSITIONShinichiMochizukiFebruary2009Wecombinevariouswell-knowntechniquesfromthetheoryofheights,thetheoryof“noncriticalBelyimaps”,andclassicalanalyticnumbertheorytocon-cludethatthe“ABCConjecture”,or,equivalently,theso-

2、called“EffectiveMordellConjecture”,holdsforarbitraryrationalpointsoftheprojectivelineminusthreepointsifandonlyifitholdsforrationalpointswhicharein“sufficientlygeneralpo-sition”inthesensethatthefollowingpropertiesaresatisfied:(a)therationalpointunderconsiderationis

3、boundedawayfromthethreepointsatinfinityatagivenfinitesetofprimes;(b)theGaloisactiononthel-powertorsionpointsofthecorrespondingellipticcurvedeterminesasurjectionontoGL2(l),forsomeprimenumberlwhichisroughlyoftheorderofthesumoftheheightoftheellipticcurveandtheloga

4、rithmofthediscriminantoftheminimalfieldofdefinitionoftheellipticcurve,butdoesnotdividetheconductoroftheellipticcurve,therationalprimesthatareabsolutelyramifiedintheminimalfieldofdefinitionoftheellipticcurve,orthelocalheights[i.e.,theordersoftheq-parameteratprimesof

5、[bad]multiplicativereduction]oftheellipticcurve.IntroductionIntheclassicalintersectiontheoryofsubvarieties,orcycles,onalgebraicvari-eties,variousversionsofthe“movinglemma”allowonetoreplaceagivencyclebyanothercyclewhichisequivalent,fromthepointofviewofintersect

6、iontheory,tothegivencycle,butissupportedonsubvarietieswhichareina“moreconvenient”position—i.e.,typically,a“moregeneral”position,whichisfreeofinessential,exceptionalpathologies—withintheambientvariety.Inthepresentpaper,afterreviewing,partlyforthepurposeofestabl

7、ishingnotationandterminology,thegeneraltheoryofheightsin§1,weproceedin§2toapplythetheoryof“noncriticalBelyimaps”of[Mzk1],togetherwithatechniquedevelopedbyElkies[cf.[Elkies];[vF]]—which,infact,maybetracedbacktotheworkofMoret-BaillyandSzpiro[cf.thediscussionatth

8、etopof[Elkies],p.106]—forrelatingBelyimapstotheABCConjecture,toshowthattheABC,EffectiveMordell,andVojtaConjecturesforarbitraryrationalpointsofhyperboliccurvesovernumberfieldsareequiv