An Introduction to p-adic Teichmuller Theory

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1、AnIntroductiontop-adicTeichm¨ullerTheorybyShinichiMochizukiThegoalofthepresentmanuscriptistoprovideanintroductiontothetheoryofuniformizationofp-adichyperboliccurvesandtheirmoduliof[Mzk1,2].Ontheonehand,thistheorygeneralizestheFuchsianandBersuniformizationsofcomplex

2、hyperboliccurvesandtheirmodulitononarchimedeanplaces.Itisforthisreasonthatweshalloftenrefertothistheoryasp-adicTeichm¨ullertheory,forshort.Ontheotherhand,thetheoryunderdiscussionmayberegardedasafairlyprecisehyperbolicanalogueoftheSerre-Tatetheoryofordinaryabelianva

3、rietiesandtheirmoduli.§1.FromtheComplexTheorytothe“ClassicalOrdinary”p-adicTheoryInthis§,weattempttobridgethegapforthereaderbetweentheclassicaluniformiza-tionofahyperbolicRiemannsurfacethatonestudiesinanundergraduatecomplexanalysiscourseandthepointofviewespousedin[

4、Mzk1,2].§1.1.TheFuchsianUniformizationLetXbeahyperbolicalgebraiccurveoverC,thefieldofcomplexnumbers.Bythis,wemeanthatXisobtainedbyremovingrpointsfromasmooth,proper,connectedalgebraiccurveofgenusg(overC),where2g−2+r>0.Weshallreferto(g,r)asthetypeofX.Thenitiswell-know

5、nthattoX,onecanassociateinanaturalwayaRiemannsurfaceXwhoseunderlyingpointsetisX(C).WeshallrefertoRiemannsurfacesXobtainedinthiswayas“hyperbolicoffinitetype.”Nowperhapsthemostfundamentalarithmetic–read“arithmeticattheinfiniteprime”–factknownaboutthealgebraiccurveXisth

6、atXadmitsauniformizationbytheupperhalfplaneH:H→XForconvenience,weshallrefertothisuniformizationofXinthefollowingastheFuch-sianuniformizationofX.Putanotherway,theuniformizationtheoremquotedaboveassertsthattheuniversalcoveringspaceXofX(whichitselfhasthenaturalstruct

7、ureofaRiemannsurface)isholomorphicallyisomorphictotheupperhalfplaneH={z∈C

8、Im(z)>0}.Thisfactwas“familiar”tomanymathematiciansasearlyasthemid-nineteenthcentury,butwasonlyprovenrigorouslymuchlaterbyK¨obe.1Thefundamentalthrustof[Mzk1,2]istogeneralizetheFuchsianuniformi

9、zationtothep-adiccontext.UpperHalfPlaneRiemannSurfaceFig.1:TheFuchsianUniformizationAtthispoint,thereadermightbemovedtointerject:Buthasn’tthisalr