correspondences on hyperbolic curves

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1、CorrespondencesonHyperbolicCurvesbyShinichiMochizuki§0.IntroductionThepurposeofthispaperistoproveseveraltheoremsconcerningthefinitenessand,moregenerally,thescarcityofcorrespondencesonhyperboliccurvesincharacteristiczeroandtocommentonthemeaningoftheseresults,especiallyrelativetotheanalogywithabelian

2、varieties.Weconsiderhyperboliccurvesoveranalgebraicallyclosedfieldkofcharacteristiczero.WecalltwosuchcurvesX,YisogenousifthereexistsanonemptyschemeC,togetherwithfinite´etalemorphismsC→X,C→Y.(Werefertosuchapair(C→X,C→Y)asacorrespondencefromXtoY.)Itiseasytoseethattherelationofisogenyisanequivalencerel

3、ationonthesetofisomorphismclassesofhyperboliccurvesoverk.Thenthefirstmainresultofthispaper(cf.Lemma4.1andTheorem4.2inthetext)isthefollowing:TheoremA.Letkbeanalgebraicallyclosedfieldofcharacteristiczero.LetXbeahyperboliccurveoverk.Let(g,r)beapairofnonnegativeintegerssatisfying2g−2+r>0.Then(uptois

4、omorphism)thereareonlyfinitelymanyhyperboliccurvesoverkoftype(g,r)thatareisogenoustoX.Moreover,ifKisanalgebraicallyclosedfieldextensionofk,thenanycurvewhichisisogenoustoXoverKisdefinedoverkandalreadyisogenoustoXoverk.Thisresultis,technicallyspeaking,arathertrivialconsequenceofhighlynontrivialresult

5、sofMargulisandTakeuchi([Marg],[Take]).Moreover,itispossiblethatTheoremAhasbeenknowntomanyexpertsforsometime,butthattheysimplyneverbotheredtowriteitdown.Asfortheauthor,IwasdimlyawareofTheoremAforsometime,withouthavingcheckedthedetailsoftheproofofit,untilIwasaskedexplicitlyaboutthefinitenessstatedinT

6、heoremAbyProf.FransOortduringmystayatUtrechtUniversityinNovember1996.IwasthenencouragedbyProf.Oorttowritedownthedetails;whencethepresentpaper.Infact,forgeneralcurves,wecansaymore:Indeed,let(Mg,r)kdenotethemodulistackofr-pointedsmooth(proper)curvesofgenusg.Here,thermarkedpointsareunordered.(Notetha

7、tthisdiffersslightlyfromtheusualconvention.)Thecomplementofthedivisorofmarkedpointsofsuchacurvewillbeahyperboliccurveoftype(g,r).Thus,weshallalsorefer(byslightabuseofterminology)to(Mg,r)kasthemodulistackofhyperbol