the grothendieck conjecture on the fundamental groups of algebraic curves

《the grothendieck conjecture on the fundamental groups of algebraic curves》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、TheGrothendieckConjectureontheFundamentalGroupsofAlgebraicCurvesHiroakiNakamura,AkioTamagawa,ShinichiMochizukiThe“GrothendieckConjecture”inthetitleis,inaword,aconjecturetotheeffectthatthearithmeticfundamentalgroupofahyperbolicalgebraiccurvecompletelydeterminesthealgebraics

2、tructureofthecurve.Researchconcerningthisproblemwasbegunattheendofthe1980’sbythefirstauthor(Nakamura),givensignificantimpetus(includingthecaseofpositivecharacteristic)bythesecondauthor(Tamagawa),andbroughttoafinalsolutionbymeansofanewp-adicinterpretationoftheproblemduetothet

3、hirdauthor(Mochizuki).Inthispaper,afterbrieflyreviewingthebackgroundandhistoryoftheproblem,wewouldliketoreportonhowtheConjecturewasgraduallybroughttoasolutionbytheworkofthethreeauthors.§1.TheArithmeticFundamentalGroup—aBridgebetweenAlgebraicGeometryandGroupTheory—§1.1.TheE

4、taleFundamentalGroup´Asiswell-known,theusual“topologicalfundamentalgroup”isaso-calledhomotopyinvari-ant,i.e.,invariantwithrespecttocontinuousdeformationsofshape.Forinstance,inthecaseofacompactcomplexalgebraiccurve,theonlyinvariantofthecurvedeterminedbyitstopo-logicalfunda

5、mentalgroupisitsgenus.Thus,takenalone,thetopologicalfundamentalgroupcannotpossiblybeasufficientlyfineinvarianttodistinguishthealgebraicstructureofdifferentalgebraiccurves.Indeed,the“arithmeticfundamentalgroup”appearingintheGrothendieckConjectureisanotionwhichisnaturallydefined

6、—asanextensionofthenotionof“Galoisgroup”—bymeansofthenotionof“´etale(i.e.,asopposedtotopological)fundamentalgroup”introducedbyA.Grothendieck.Thisnotionof“´etalefundamentalgroup”wasintroducedintoalgebraicgeometryinthe1960’sin[SGA1]asanaccountingdevicetokeeptrackofthe“Galoi

7、stheoryofschemes.”Accordingto[SGA1],givenageometricpoint¯xonaconnectedschemeX,the´etalefundamentalgroupπ1(X,x¯)isdefinedasagroupofpermutationsofasystemof“setsofsolutions”asfollows:AsYrangesoverallofthefinite´etalecoverings(inthefollowing,weshallfrequentlyabbreviatethisexpre

8、ssionbythephrase“finitecoverings”)ofX,thefibersetsY(¯x)overthegeometricpoint¯xformaprojectivesyste