DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(3)

Let #(u) and #(u) be the usual functions in the theory of elliptic functions. The following two formulae were found in the nineteenth-century. First one is

HYPERELLIPTICFUNCTIONSINGENUSTHREE3

withappropriatechoiceofapathoftheintegrals.Needlesstosay,wehave(x(0,0,0),y(0,0,0))=∞.

Fromourstandingpointofview,thefollowingthreefeaturesstandoutontheformulae(1.3)and(1.4).Firstly,thesequenceoffunctionsofuwhosevaluesatu=ujaredisplayedinthe(j+1)-throwofthedeterminantof(1.3)isasequenceofthemonomialsofx(u)andy(u)displayedaccordingtotheorderoftheirpolesatu=0.Secondly,whiletherighthandsidesof(1.3)and(1.4)arepolynomialsofx(u)andy(u),whereu=u0for(1.4),thelefthandsidesareexpressedintermsofthetafunctions,whosedomainisproperlytheuniversalcoveringspaceC(oftheJacobianvariety)oftheellipticcurve.Thirdly,theexpressionofthelefthandsideof(1.4)statesthefunctionofthetwosidesthemselvesof(1.4)ischaracterizedasahyperellipticfunctionsuchthatitszeroesareexactlythepointsdi erentfrom∞whosen-plicationisjustonthestandardthetadivisorintheJacobianofthecurve,andsuchthatitspoleisonlyat∞.Inthecaseoftheellipticcurveabove,thestandardthetadivisorisjustthepointatin nity.

Surprisinglyenough,thesethreefeaturesjustinventgoodgeneralizationsforhyperellipticcurves.Moreconcreatly,ourgeneralizationof(1.4)isobtainedbyreplacingthesequenceoftherighthandsidebythesequence

1,x(u),x2(u),x3(u),y(u),x4(u),yx(u),···,

whereu=(u(1),u(2),u(3))isonκ 1ι(C),ofthemonomialsofx(u)andy(u)dis-playedaccordingtotheorderoftheirpolesatu=(0,0,0)withreplacingthederivativeswithrespecttou∈Cbythosewithrespecttou(1)alongκ 1ι(C);andthelefthandsideof(1.4)by

1!2!···(n 1)!σ(nu)/σ2(u)n,

whereσ(u)=σ(u(1),u(2),u(3))isawell-tunedRiemannthetaseriesandσ2(u)=( σ/ u(2))(u).Therefore,thehyperellipticfunctionthatistherighthandsideofthegenerlarizationof(1.4)isnaturallyextendedtoafunctiononC3viathetafunctions.AlthoughtheextendedfunctiononC3isnolongerafunctionontheJacobian,itisexpressedsimplyintermsofthetafunctionsandistreatedreallysimilarwaytotheellipticfunctions.Themostdi cultproblemisto ndthelefthandsideoftheexpectedgeneralizationof(1.3).Theanswerisremarkablyprettyandis σ(u0+u1+···+un)i<jσ3(ui uj)2