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1、CorrespondencesonHyperbolicCurvesbyShinichiMochizuki§0.IntroductionThepurposeofthispaperistoproveseveraltheoremsconcerningthe?nitenessand,moregenerally,thescarcityofcorrespondencesonhyperboliccurvesincharacteristiczeroandtocommentonthemeaningoftheseresults,especiallyrelativetotheanalogywit
2、habelianvarieties.Weconsiderhyperboliccurvesoveranalgebraicallyclosed?eldkofcharacteristiczero.WecalltwosuchcurvesX,YisogenousifthereexistsanonemptyschemeC,togetherwith?nite′etalemorphismsC→X,C→Y.(Werefertosuchapair(C→X,C→Y)asacorrespondencefromXtoY.)Itiseasytoseethattherelationofisogenyis
3、anequivalencerelationonthesetofisomorphismclassesofhyperboliccurvesoverk.Thenthe?rstmainresultofthispaper(cf.Lemma4.1andTheorem4.2inthetext)isthefollowing:TheoremA.Letkbeanalgebraicallyclosed?eldofcharacteristiczero.LetXbeahyperboliccurveoverk.Let(g,r)beapairofnonnegativeintegerssatisfyi
4、ng2g?2+r>0.Then(uptoisomorphism)thereareonly?nitelymanyhyperboliccurvesoverkoftype(g,r)thatareisogenoustoX.Moreover,ifKisanalgebraicallyclosed?eldextensionofk,thenanycurvewhichisisogenoustoXoverKisde?nedoverkandalreadyisogenoustoXoverk.Thisresultis,technicallyspeaking,arathertrivialcon
5、sequenceofhighlynontrivialresultsofMargulisandTakeuchi([Marg],[Take]).Moreover,itispossiblethatTheoremAhasbeenknowntomanyexpertsforsometime,butthattheysimplyneverbotheredtowriteitdown.Asfortheauthor,IwasdimlyawareofTheoremAforsometime,withouthavingcheckedthedetailsoftheproofofit,untilIwasa
6、skedexplicitlyaboutthe?nitenessstatedinTheoremAbyProf.FransOortduringmystayatUtrechtUniversityinNovember1996.IwasthenencouragedbyProf.Oorttowritedownthedetails;whencethepresentpaper.Infact,forgeneralcurves,wecansaymore:Indeed,let(Mg,r)kdenotethemodulistackofr-pointedsmooth(proper)curvesofg
7、enusg.Here,thermarkedpointsareunordered.(Notethatthisdi?ersslightlyfromtheusualconvention.)Thecomplementofthedivisorofmarkedpointsofsuchacurvewillbeahyperboliccurveoftype(g,r).Thus,weshallalsorefer(byslightabuseofterminology)to(Mg,r)kasthemodulistackofhyperbol
