correspondences on hyperbolic curves

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

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

3、anequivalencerelationonthesetofisomorphismclassesofhyperboliccurvesoverk.Thenthefirstmainresultofthispaper(cf.Lemma4.1andTheorem4.2inthetext)isthefollowing:TheoremA.Letkbeanalgebraicallyclosedfieldofcharacteristiczero.LetXbeahyperboliccurveoverk.Let(g,r)beapairofnonnegativeintegerssatisfyi

4、ng2g−2+r>0.Then(uptoisomorphism)thereareonlyfinitelymanyhyperboliccurvesoverkoftype(g,r)thatareisogenoustoX.Moreover,ifKisanalgebraicallyclosedfieldextensionofk,thenanycurvewhichisisogenoustoXoverKisdefinedoverkandalreadyisogenoustoXoverk.Thisresultis,technicallyspeaking,arathertrivialcon

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

6、skedexplicitlyaboutthefinitenessstatedinTheoremAbyProf.FransOortduringmystayatUtrechtUniversityinNovember1996.IwasthenencouragedbyProf.Oorttowritedownthedetails;whencethepresentpaper.Infact,forgeneralcurves,wecansaymore:Indeed,let(Mg,r)kdenotethemodulistackofr-pointedsmooth(proper)curvesofg

7、enusg.Here,thermarkedpointsareunordered.(Notethatthisdiffersslightlyfromtheusualconvention.)Thecomplementofthedivisorofmarkedpointsofsuchacurvewillbeahyperboliccurveoftype(g,r).Thus,weshallalsorefer(byslightabuseofterminology)to(Mg,r)kasthemodulistackofhyperbol