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MEASURESOFSIMULTANEOUSAPPROXIMATIONFORQUASI-PERIODSOFABELIANVARIETIESPIERREGRINSPAN1.IntrodutionThistexthasitssoureinseveral,botholdandreent,results: Firstandforemost,aseriesoftheoremsofChudnovskyfromthe70’s([Chu84℄,Chap.7)whih,astheyareentralhereandsomewhatsatteredinChudnovsky’sbook,wepresentlyreall:Theorem1.1(Chudnovsky’stheorems).Let =Z!+Z!0 Cbealattiewithinvariantsg2,g3,Weierstrassfuntions}, andquasi-periods , 0,allde nedintheusualway(see[Sil94℄).Eahofthesetsbelowontainsatleasttwoalgebraiallyindependentnumbers: rst(1)fg2;g3; !; !g,(2)fg2;g3;!; ;!0; 0g;ifweassumeg2;g32 Q:(3)f !; !g;(4)f!;!0; ; 0g;andif,stillassumingg2;g32 Q,weonsidertwoomplexnumbersuandu0,Q-linearlyindependentandsuhthat}(u);}(u0)2 Q:(5)f !; (u) !ug;(6)fu;u0; (u); (u0)g:Itanbenotiedthattheaboveassertionsarerelated,regard-lessoftheirtruthfulness,bythefollowinglogialimpliations:(1))(3)((5)++(2))(4)((6) Seond,aresultannounedinChudnovsky’sbook([Chu84℄,The-orem9p.9)whihextendsassertion(4)abovetoabelianvarietiesofarbitrarydimension;aompleteproofwasreentlygivenin[Vas96℄.1 Third,Theorem4.1of[RW97℄,whereameasureofsimultane-ousapproximationisestablishedwhihhasTheorem1.1(4)asaorollary. Fourth,a\trikintroduedbyChudnovskyin[Chu82℄toproveasharpmeasureofalgebraiindependenere ningassertion(3)above;reentlyredisoveredin[Phi99℄,[Bru99℄,itonsistsinre-latingelliptiandquasi-ellipti(likeWeierstrass’s )funtionstoG-funtions,allowingbetterarithmetiestimatesinthetransen-deneproofand,ultimately,anoptimaldependeneofmeasuresintheparameterontrollingtheheight.Oneimportantfeatureofourresults,omingfrompointThreeabove,isthefollowing:De nition1.1.A(simultaneous)approximationmeasurefor( 1;:::; n)2Cnisafuntion :N R+!R+suhthatforsomeonstantC0,forany( 1;:::; n)2 Qnwith[Q( 1;:::; n):Q℄ D,h( i) h(absolutelogarithmiheightasde nedin[Wal92℄)andD;h Conehaslogmaxij i ij C (D;h):Beforestatingourmainresult,wereallafewmorede nitions.LetAbeanabelianvarietyofdimensiongde nedoverasub eldKofC.Arational(henemeromorphi)di erentialonAissaidtobeoftheseondkindifithasnoresidues([GH78℄,p.454);thequotientspaeofseond-kindbyexatdi erentialshasdimension2gandwillbedenotedbyH1DR(A;K),orH1DR(A)whenweimpliitlytakeK=C(thenotationH1DRisjusti edby[FW84℄,p.192).Ontheotherhand,H1(A;Z)willdenotetheusual rsthomologygroupofA(C).Theorem1.2.LetAbeanabelianvarietyde nedovera eldK C,(!1;:::;!2g)representingabasisofH1DR(A;K)andu1;:::;ur2T0A(C)(tangentspaeattheorigin),Q-linearlyindependentandsuhthatexpA(uj)2A(K).Welet =rg.1.IfK Q,thesetofRuj0!i(1 i 2g;1 j r)admitsthefollowingapproximationmeasure: 1(D;h)=D32+1 (logD) 1=2(D2= +h):2.Ifalltheujareperiods(elementsoftheperiodlattie =kerexpA),thesetmadeupbyallRuj0!i(1 i 2g;1 j r)togetherwithageneratingsystemofKoverQadmitsthefollow-ingapproximationmeasure: 2(D;h)=[D(h+logD)℄3= :2UsingatheoremofLaurent-Roy([LR99℄,Th eor eme1.2)weande-duefromassertions(1)and(2)(resp.)ofthepreedingtheoremexten-sionsinarbitrarydimensionofassertions(6)and(2)(resp.)ofTheorem1.1:Corollary1.1.1.IfK Qandr=2g,theRuj0!igeneratea eldoftransendenedegreeatleast2.2.Ifr g+1andtheujareperiods,the eldgeneratedoverKbytheRuj0!ihastransendenedegree(overQ)atleast2.Thetextisarrangedasfollows.Setion2brie yreviewsembed-dingsofextensionsofabelianvarietiesbypowersoftheadditivegroup[FW84℄,andendswithazeroestimate,orollaryof[Phi96℄,tailoredtothispartiulartypeofalgebraigroups.Setion3ontainsaddition,multipliationanddi erentiationformulaeforthefuntionsinvolvedintheseembeddings,muhinthespiritandontinuationof[MW93℄x3;therewealsoonstrut,startingfromlassialthetafuntions,\sigmafuntionswhiharenothingbutthe\algebraithetafuntionswhoseexisteneisprovedin[Bar70℄.Thenextsetion(x4)develops,intheontextofthealgebraigroupsdesribedabove,Chudnovsky’s\G-trikmentionedearlier(pointFour);itisbasedonaquitegeneralande etiveversionofEisenstein’slassialtheoremstatingthateveryalgebraipowerseriesisaG-funtion(see[PS76℄,VIII.3.3&VIII.4.4).Insetionx5,wereviewbrie ytheveryspeialbutlassialaseg=1ofthegeneralresultsontainedintheprevioustwosetions.Insetion6westateinfulldetailandarryouttheproofofourmainresult;andinsetion7,wedisussvariousresultsloselyrelatedtoourmaintheorem,statesomewithsummaryindiationsofproofs,andexam-inewhihtehnialdiÆulties(omingfromtheinsuÆienyofknownShwarz/interpolationlemmas)ariseinprovingmore.2.EmbeddingsandzerolemmaHerewewilldesribethetypeofembeddingswewillbeusingforextensions,bypowersoftheadditivegroupGa,ofprinipallypolarizedabelianvarieties.LetAbesuhanabelianvariety, CalattiesuhthatA(C)’C= .Fori=1:::gwritei=zi,andforanyderivation,logf=ff.Thefollowing,elementarybutfundamental,lemmaprovidesuswithanexpliitbasisforthequotientH1DR(A;C)ofthespaeof rst-ordermeromorphidi erentialsoftheseondkindonA(i.e.withoutresidues)bythatofexatdi erentials.Wereferthereaderto[Lan82℄3forboththede nitionofanondegeneratethetafuntion(allofthoseonsideredinthistextwillbeso)andtheredutionproessusedintheproof,astheybotharelassialandwillnotbeusedintherestofthetext.Lemma2.1.Foranynondegeneratethetafuntion forthelattie ,thedi erentialformsdz1;:::;dzg(oordina