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1、CorrespondencesonHyperbolicCurvesbyShinichiMochizuki§0.IntroductionThepurposeofthispaperistoproveseveraltheoremsconcerningthe?nitenessand,moregenerally,thescarcityofcorrespondencesonhyperboliccurvesincharacteristiczeroandtocommentonthemeaningoftheseresults,especiallyrelativetotheanalogywithabelian
2、varieties.Weconsiderhyperboliccurvesoveranalgebraicallyclosed?eldkofcharacteristiczero.WecalltwosuchcurvesX,YisogenousifthereexistsanonemptyschemeC,togetherwith?nite′etalemorphismsC→X,C→Y.(Werefertosuchapair(C→X,C→Y)asacorrespondencefromXtoY.)Itiseasytoseethattherelationofisogenyisanequivalencerel
3、ationonthesetofisomorphismclassesofhyperboliccurvesoverk.Thenthe?rstmainresultofthispaper(cf.Lemma4.1andTheorem4.2inthetext)isthefollowing:TheoremA.Letkbeanalgebraicallyclosed?eldofcharacteristiczero.LetXbeahyperboliccurveoverk.Let(g,r)beapairofnonnegativeintegerssatisfying2g?2+r>0.Then(uptois
4、omorphism)thereareonly?nitelymanyhyperboliccurvesoverkoftype(g,r)thatareisogenoustoX.Moreover,ifKisanalgebraicallyclosed?eldextensionofk,thenanycurvewhichisisogenoustoXoverKisde?nedoverkandalreadyisogenoustoXoverk.Thisresultis,technicallyspeaking,arathertrivialconsequenceofhighlynontrivialresult
5、sofMargulisandTakeuchi([Marg],[Take]).Moreover,itispossiblethatTheoremAhasbeenknowntomanyexpertsforsometime,butthattheysimplyneverbotheredtowriteitdown.Asfortheauthor,IwasdimlyawareofTheoremAforsometime,withouthavingcheckedthedetailsoftheproofofit,untilIwasaskedexplicitlyaboutthe?nitenessstatedinT
6、heoremAbyProf.FransOortduringmystayatUtrechtUniversityinNovember1996.IwasthenencouragedbyProf.Oorttowritedownthedetails;whencethepresentpaper.Infact,forgeneralcurves,wecansaymore:Indeed,let(Mg,r)kdenotethemodulistackofr-pointedsmooth(proper)curvesofgenusg.Here,thermarkedpointsareunordered.(Notetha
7、tthisdi?ersslightlyfromtheusualconvention.)Thecomplementofthedivisorofmarkedpointsofsuchacurvewillbeahyperboliccurveoftype(g,r).Thus,weshallalsorefer(byslightabuseofterminology)to(Mg,r)kasthemodulistackofhyperbol
