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ID:15176606
大小:648.44 KB
页数:97页
时间:2018-08-01
《combinatorial anabelian topics iii》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、TOPICSSURROUNDINGTHECOMBINATORIALANABELIANGEOMETRYOFHYPERBOLICCURVESIII:TRIPODSANDTEMPEREDFUNDAMENTALGROUPSYUICHIROHOSHIANDSHINICHIMOCHIZUKIFEBRUARY2013Abstract.LetΣbeasubsetofthesetofprimenumberswhichiseitherequaltotheentiresetofprimenumbersorofcardinal-ityone.Inthepresentpaper,w
2、econtinueourstudyofthepro-Σfundamentalgroupsofhyperboliccurvesandtheirassociatedcon-figurationspacesoveralgebraicallyclosedfieldsinwhichtheprimesofΣareinvertible.Thefocusofthepresentpaperisonappli-cationsofthetheorydevelopedinpreviouspaperstothetheoryoftemperedfundamentalgroups,inth
3、estyleofAndr´e.Theseapplicationsaremotivatedbythegoalofsurmountingtwofunda-mentaltechnicaldifficultiesthatappearinpreviousworkofAndr´e,namely:(a)thefactthatthecharacterizationofthelocalGaloisgroupsintheglobalGaloisimageassociatedtoahyperboliccurvethatisgiveninearlierworkofAndr´eison
4、lyprovenforaquitelimitedclassofhyperboliccurves,i.e.,aclassthatis“farfromgeneric”;(b)theproofgiveninearlierworkofAndr´eofacertainkeyinjectivityresult,whichisofcentralimportanceinestablishingthetheoryofa“p-adiclocalanalogue”ofthewell-known“global”theoryoftheGrothendieck-Teichm¨ulle
5、rgroup,containsafundamentalgap.Inthepresentpaper,wesurmountthesetechnicaldifficultiesbyintroduc-ingthenotionofan“M-admissible”,or“metric-admissible”,outerautomorphismoftheprofinitegeometricfundamentalgroupofap-adichyperboliccurve.Roughlyspeaking,M-admissibleouterautomorphismsareouter
6、automorphismsthatarecompatiblewiththedataconstitutedbytheindicesatthevariousnodesofthespe-cialfiberofthep-adiccurveunderconsideration.Bycombiningthisnotionwithcombinatorialanabelianresultsandtechniquesde-velopedinearlierpapersbytheauthors,togetherwiththetheoryofcyclotomicsynchroniz
7、ation[alsodevelopedinearlierpapersbytheauthors],weobtainageneralizationofAndr´e’scharacterizationofthelocalGaloisgroupsintheglobalGaloisimageassociatedtoahyperboliccurvetothecaseofarbitraryhyperboliccurves[cf.(a)].Moreover,byapplyingthetheoryoflocalcontractibilityofp-adicanalytics
8、pacesdevelopedbyBerkovich,weshowt
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