On the structure of the class group of a function field

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1、Vol.33,197933OnthestructureoftheclassgroupofafunctionfieldVonG~DFRay1)1.Introduction.Let2'beafunctionfieldofonevariabledefinedoverafieldKwithconservativegenusg.LetTbethecharacteristicofK,_~thealgebraicclosureofK.TheclassgroupC(~V)consistsofthedivisorclassesofdegree0.Forme~wedefin

2、e:C(F)m={ceC(~);m.c=0}2),C(F)(m)----(ceC(F),3keNwithms.c:0}.ForKcLwedenotewith-~.Lthefunctionfieldobtainedbyconstantfieldex-tensionfromKtoL.Thefollowingtheoremiswellknown:Theorem1.I/meNandp~mthenC(F.E)m_~(Z/m)2g.IntheusualproofofthisfactoneusestheJacobianofFthatisanabelianvariety

3、AofJimensiong,andthenoneprovesthatA(K)m~---(Z/m)2nforallabe]Janva-rietiesofdimensionn.TogetthisfactonecanconsiderabelianvarietiesoverCasanalytictoritoprovetheresultforK=C,andtherestisreductiontheory(cf.[7]).Anotherpossibilitythatispurelygeometricisshownin[6].Herewewanttogiveanele

4、mentaryproofthatusesonlythetheoremofRiemann-Rochandthe"easy"partofthereductiontheory.Inw2welookatthespecialcasethatKisafinitefield.ThentheelementarypropertiesoftheZeta-functionof-~(thatareobtainedbyapplicationoftheRiemann-Roch-theorem)andthesurjeetivityofthenormmapwithrespecttofi

5、niteconstantfieldextensionsgivethetheorem.Inw4weusethereductionmapinthecaseofgoodreductioninordertogetthatforanyfieldKwehaveIC,(F.~)m]~m2g-Themostdelicatestepisdoneinw3:Wehavetoliftelementsofordermoftheclassgroupofthereducedfunctionfieldintotheclassgroupofafunctionfieldoveralocal

6、field.(Thisfieldplaysthe1)ThispaperwaspartlywrittenwhentheauthorwasvisitingtheOhioStateUniversity,Columbus,Ohio.2)Thelawofcompositioninthedivisorgroupandinthedivisorclassgroupiswrittenad-ditively.ArchlyderMathematik33334G.F~YARCH.MATH.roleofthecomplexnumbershere.)Wecoulduseageome

7、tricargumentinvolvingdeformationtheoryandformalgeometrythatassuresthatanyuuramifiedextensionofafunctionfieldcanbelifted(andelementsofordermintheclassgroupcorrespondtocyclicunramifiedextensionsofdegreemviaKummertheory)(cf.[7])butthiswouldbetoocomplicatedcomparedwiththesimplicityof

8、theconstructionoftheJaeobianoverCthatwew