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1、MATHEMATICSOFCOMPUTATIONVolume66,Number219,July1997,Pages1161{1168S0025-5718(97)00860-0UNIVERSALBINARYHERMITIANFORMSA.G.EARNESTANDAZARKHOSRAVANIAbstract.Wewilldetermine(uptoequivalence)alloftheintegralpositivede niteHermitianlatticesinimaginaryquadratic eldsofclassnumber1thatr
2、epresentallpositiveintegers.1.IntroductionThesearchforpositivede nitequaternaryintegralquadraticformswhichrepre-sentallpositiveintegershasalongandillustrioushistory,datingbacktoLagrange'sproofin1770thattheformx2+y2+z2+w2hasthisproperty.Suchformsarereferredtoasuniversalinthecon
3、temporaryliterature.Moregenerally,apositiveintegralquadraticformoveratotallyrealnumber eldissaidtobeuniversalifeverytotallypositiveintegerofthe eldisrepresentedbytheform.Whilenouni-versalpositivebinaryquadraticformsexist,Maass[8]showedthatthesumofthreepsquaresisuniversaloverQ(
4、5).In1994,Chan,KimandRaghavan[1]showedthatpppamongtherealquadratic elds,onlythe eldsQ(2),Q(3),andQ(5)admituniversalternaryclassicintegralquadraticforms;allsuchformsarelistedbytheauthors.Inthispaper,weconsidertheanalogousproblemof ndinguniversalpositivede niteHermitianforms.Itw
5、illbeshownthatoverallimaginaryquadratic elds,thereexistonly nitelymanyclassesofuniversalbinarypositivede niteHermit-ianforms.Allsuchformswillbedeterminedfortheimaginaryquadratic eldspofclassnumber1;i.e.,the eldsQ(m)wherem=−1;−2;−3;−7;−11;−19;−43;−67;−163.Computationalmethodswe
6、reusedtoproducealistcontainingallpotentiallyuniversalbinaryHermitianforms,andallclassesintheirgenera,overthenineimaginaryquadratic eldsofclassnumber1.Wenowgiveabriefoutlineofthemethodused.First,anupperboundforthediscriminantofauniversalbinaryHermitianformisdeterminedforeachoft
7、he elds.Next,inequalitiesareobtainedviareductiontheoryforthecoecientsofsuchforms.Thepotentiallyuniversalreducedformsarethenlistedandarescreenedforpossibleuniversalitybydetermin-ingwhethertheintegers1through5arerepresented.Thisroughscreeningleavesthirteencandidates.Ofthese,six
8、giveriseviathetracemappingofJacobson[4]todiagonalintegralquat
