DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(12)

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

12 YOSHIHIROONISHI

Theorem3.2.Letn≥2beaninteger.Assumethatu0,u1,···,unbelongtoκ 1ι(C).Then σ(u0+u1+···+un)i<jσ3(ui uj)

σ2(u)3σ2(u1)3σ2(u2)3

Proof.Wesupposethatu,u1,u2areanypointsnotonκ 1ι(C).Sincethesumof

pull-backsoftranslationsTuΘ+T u1Θ+T u2Θislinearlyequivalentto3Θ1+u2

bythetheoremofsquare([11],Coroll.4inp.59),thefunction

σ(u+u1+u2)σ(u u1)σ(u u2)σ(u1 u2) 1 = 1 1x(u)x(u1)x(u2) x(u) x2(u1) .x2(u2) 2

33(u u1)

22 333 222 33 22(u1 u2)(u)(u2)

tothefunctionabove,bybringingu,u1,andu2closetopointsonκ 1ι(C),wehavethelefthandsideoftheclaimedfurmula.Herewehaveusedthefactthatσ(u u1),σ(u u2),andσ(u1 u2)vanishforu,u1,andu2onκ 1ι(C)byLemma

2.4(2).Sothelefthandsideasafunctionofuonκ 1ι(C)isperiodicwithrespect