noncritical belyi maps

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1、NoncriticalBelyiMapsShinichiMochizukiMay2004Inthepresentpaper,wepresentaslightlystrengthenedversionofawell-knowntheoremofBelyiontheexistenceof“Belyimaps”.Roughlyspeaking,thisstrengthenedversionassertsthatthereexistBelyimapswhichareunramifiedat[cf.Theorem2.5]—orevenne

2、ar[cf.Corollary3.2]—aprescribedfinitesetofpoints.Section1:IntroductionWriteCforthecomplexnumberfield;Q⊆Cforthesubfieldofalgebraicnumbers.LetXbeasmooth,proper,connectedalgebraiccurveoverQ.IfFisafield,thenweshalldenotebyP1theprojectivelineoverF.FDefinition1.1.Weshallrefertoadominan

3、tmorphism[ofQ-schemes]1φ:X→PQasaBelyimapifφisunramifiedovertheopensubschemeU⊆P1givenbythePQcomplementofthepoints“0”,“1”,and“∞”ofP1;inthiscase,weshallrefertoQdef−1UX=φ(UP)⊆XasaBelyiopenofX.In[1],itisshownthatXalwaysadmitsatleastoneBelyiopen.Fromthispointofview,themainresult(Th

4、eorem2.5)ofthepresentpaperhasasanimmediateformalconsequence(pointedouttotheauthorbyA.Tamagawa)thefollowinginteresting[andrepresentative]result:Corollary1.2.(BelyiOpensasaZariskiBase)IfVX⊆XisanyopensubschemeofXcontainingaclosedpointx∈X,thenthereexistsaBelyiopenUX⊆VX⊆Xsuchthat

5、x∈UX.Inparticular,theBelyiopensofXformabasefortheZariskitopologyofX.Acknowledgements:TheauthorwishestothankA.TamagawaforhelpfuldiscussionsduringNo-vember1999concerningtheproofofTheorem2.5givenhere.TypesetbyAMS-TEX12SHINICHIMOCHIZUKISection2:TheMainResultWebeginwithsomeelemen

6、tarylemmas:Lemma2.1.(SeparatingPropertiesofBelyiMaps)LetC∈RbesuchthatC≥2;let1S⊆P(Q)beafinitesetofrationalpointssuchthat:(i)0,1,∞∈S;(ii)thereexistsanr∈Ssuchthat01.Supposethatβ∈QSsatisfiesthefollowingcondition:(iv)β/α≥C,forallα∈S{0

7、,∞}.Writer=m/(m+n),wherem,n≥1areintegers.Thenthefunctiondefmnf(x)=x·(x−1)satisfiesthefollowingproperties:(a)f({0,r,1,∞})⊆{0,f(r),∞};(b)f(x)=0(wherex∈C)impliesx∈{0,r,1,∞}⊆S;(c)f(β)∈/f(S);(d)(f(β)+f0)/(f(α)+f0)≥Cforallα∈S{∞}suchthatf(α)+f0=0.defHere,wewritef0=−minα{f(α)},whe

8、reαrangesovertheelementsofS{∞}.Proof.Property(a)isimmediatefromthedefinitio