DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(13)

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE13

toΛ.Nowweregardthebothsidestobefunctionsofuonκ 1ι(C).Weseethelefthandsidehasitsonlypoleatu=(0,0,0)moduloΛby2.8(1),andhasitszeroesatu=±u1andu=±u2moduloΛby2.4(2),2.9.Theseallzeroesareoforder1by2.4(2)and2.11.ItsLaurentexpansionatu=(0,0,0)isgivenby2.8(2)and2.10asfollows:

σ3(u1+u2)σ2(u1)σ2(u2)σ3(u1 u2)

Therighthandsideis 1 1 1x(u1) x(u2) u(3)4+···).

Proof.Weknowthelefthandsideoftheclaimedformulais,asafunctionofu,aperiodicfunctionwithrespecttoΛ.Itspoleisonlyatu=(0,0,0)moduloΛandiscontributedonlybythefunctionsσ2(u)4,σ3(u u1),σ3(u u2),σ3(u u3).By2.8(2)and2.10,theorderofthepoleis4×3 3×2,thatis6.Thezeroesofthelefthandsideareatu= u1, u2,andu3moduloΛwhicharecomingfromσ(u+u1+u2+u3);andatu=u1,u2,u3whicharecomingfromσ(u u1),σ(u u2),σ(u u3),respectively.These6zeroesareoforder1by2.11.Thusweseethatthedivisorsoftwosidescoincide.Thecoe cientofleadingtermoftheLaurentexpansionofthelefthandsideis

σ(u1+u2+u2)σ2(u1)σ2(u2)σ2(u3) i<j 1 1= 1 1x(u)x(u1)x(u2)x(u3)x2(u)x2(u1)x2(u2)x2(u3)σ2(u)4σ2(u1)4σ2(u2)4σ2(u3)4 x3(u) x3(u1) .x3(u2) x3(u3)σ3(ui uj)