Abelian varieties over finite fields s_q

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1、AbelianvarietiesoverfinitefieldsFransOortMathematischInstituut,P.O.Box.80.010,NL-3508TAUtrechtTheNetherlandse-mail:oort@math.uu.nlAbstract.A.WeilprovedthatthegeometricFrobeniusπ=Faofanabelianvarietyoverafinitefieldwithq=paelementshasabsolutevalue√qforeveryembedding.T

2、.HondaandJ.TateshowedthatA7→πAgivesabijectionbetweenthesetofisogenyclassesofsimpleabelianvarietiesoverFqandthesetofconjugacyclassesofq-Weilnumbers.Higher-dimensionalvarietiesoverfinitefields,SummerschoolinG¨ottingen,June2007IntroductionWecouldtrytoclassifyisomorphi

3、smclassesofabelianvarieties.Thetheoryofmodulispacesofpolarizedabelianvarietiesanswersthisquestioncompletely.Thisisageometrictheory.Howeverinthisgeneral,abstracttheoryitisoftennoteasytoexhibitexplicitexamples,toconstructabelianvarietieswithrequiredproperties.Acoar

4、serclassificationisthatofstudyingisogenyclassesofabelianvarieties.Awonderfulandpowerfultheorem,theHonda-Tatetheory,givesacompleteclassificationofisogenyclassesofabelianvarietiesoverafinitefield,seeTheorem1.2.ThebasicideastartswithatheorembyA.Weil,aprooffortheWeilconj

5、ec-tureforanabelianvarietyAoverafinitefieldK=Fq,see3.2:thegeometricFrobeniusπAofA/Kisanalgebraicinteger√whichforeveryembeddingψ:Q(πA)→Chasabsolutevalue

6、ψ(πA)

7、=q.ForanabelianvarietyAoverK=FqtheassignmentA7→πAassociatestoAitsgeometricFrobeniusπA;theisogenyclassofAgiv

8、estheconjugacyclassofthealgebraicintegerπA,andconverselyanalgebraicintegerwhichisaWeilq-numberdeterminesanisogenyclass,asT.HondaandJ.Tateshowed.Geometricobjectsareconstructedandclassifieduptoisogenybyasimplealge-braicinvariant.Thisarithmetictheorygivesaccesstoalot

9、ofwonderfultheorems.Inthesenoteswedescribethistheory,wegivesomeexamples,applicationsandsomeopenquestions.Insteadofreadingthesenotesitismuchbettertoreadthewonderfulandclear[73].Someproofshavebeenworkedoutinmoredetailin[74].In§§1∼15materialdiscussedinthecourseisdes

10、cribed.Intheappendices§§16∼22wehavegatheredsomeinformationweneedforstatementsandproofsofthemainresult.Ihopeallrelevantnotionsandinformationneededforunder-stand