topics surrounding the anabelian geometry of hyperbolic curves

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1、TopicsSurroundingtheAnabelianGeometryofHyperbolicCurvesShinichiMochizukiSeptember2002Contents:§0.Introduction§1.TheTateConjectureasaSortofGrothendieckConjecture§1.1.TheTateConjecturefornon-CMEllipticCurves§1.2.SomePro-pGroupTheory§2.HyperbolicCurvesastheirown“AnabelianAlbaneseVarieties”§2.1.ACoroll

2、aryoftheMainTheoremof[Mzk2]§2.2.APartialGeneralizationtoFiniteCharacteristic§3.DiscreteRealAnabelianGeometry§3.1.RealComplexManifolds§3.2.FixedPointsofAnti-HolomorphicInvolutions§3.3.HyperbolicCurvesandtheirModuli§3.4.AbelianVarietiesandtheirModuli§3.5.ProfiniteRealAnabelianGeometry§4.Complementstot

3、hep-adicTheory§4.1.GoodChernClasses§4.2.TheGroup-TheoreticityofaCertainChernClass§4.3.AGeneralizationoftheMainResultof[Mzk2]Section0:IntroductionInthismanuscript,wegiveanexpositionofvariousideasandresultsrelatedtothefundamentalresultsof[Tama1-2],[Mzk1-2]concerningGrothendieck’sConjec-tureofAnabelia

4、nGeometry(whichwerefertoasthe“GrothendieckConjecture”forshort;cf.[Mzk2],Introduction,forabriefintroductiontothisconjecture).Manyoftheseideasexistedpriortothepublicationof[Tama1-2],[Mzk1-2],butwerenotTypesetbyAMS-TEX12SHINICHIMOCHIZUKIdiscussedinthesepapersbecauseoftheirratherelementarynatureandseco

5、ndaryimportance(bycomparisontothemainresultsofthesepapers).Nevertheless,itisthehopeoftheauthorthatthereaderwillfindthepresentmanuscriptusefulasasupplementto[Tama1-2],[Mzk1-2].Inparticular,wehopethatthediscussionofthepresentmanuscriptwillservetoclarifythemeaningandmotivationbehindthemainresultof[Mzk2

6、].Ourmainresultsarethefollowing:(1)In§1,wetakethereversepointofviewtotheusualone(i.e.,thattheGrothendieckConjectureshouldberegardedasasortof(anabelian)TateConjecture)andshowthatinacertaincase,theTateConjecturemayberegardedasasortofGrothendieckConjecture(cf.Theorem1.1,Corollary1.2).Inparticular,Coro

7、llary1.2isinterestinginthatitallowsonetoexpressthefundamentalphenomenoninvolvedintheTateandGrothen-dieckConjecturesusingelementarylanguagethatcan,inprinciple,beunderstoodevenbyhighschoolstudents(cf.theIntro