DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(8)

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

8 YOSHIHIROONISHI

Lemma2.5.Withthenotationabove,wehave

13(u)=x1x2x3, 23(u)= x1x2 x1x3 x3x1, 33(u)=x1+x2+x3.Foraproofofthis,see[2],p.377.Thisfactisentirelydependsonthechoiceofformsω(j)’sandη(j)’s.

Lemma2.6.Ifu=(u(1),u(2),u(3))isonκ 1ι(C),thenwehave

u(1)=1

3u(3)+(d (u(3))≥4).3

Thisismentionedin[12],Lemma2.3.2(2).Ifuisapointonκ 1ι(C)thex-andy-coordinatesofι 1κ(u)willbedenotedbyx(u)andy(u),respectively.AsisshowninLemma2.3.1of[12],forinstance,weseethefollowing.

Lemma2.7.Ifu∈κ 1ι(C)then

x(u)=1

u(3)5+(d ≥ 4).

Lemma2.8.(1)Letubeanarbitrarypointonκ 1ι(C).Thenσ2(u)is0ifandonlyifubelongstoκ 1(O).

(2)TheTaylorexpansionofthefunctionσ2(u)onκ 1ι(C)atu=(0,0,0)isoftheform

3σ2(u)= u(3)+(d (u(3))≥5).

Proof.For(1),assumethatu∈κ 1ι(C)andu∈κ 1(O).Thenwehave

σ1(u)

23(u)=x1x2x3

σ2(u)= 33(u)

σ3(u)= 13(u)

σ3(u)= 23(u)