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

6 YOSHIHIROONISHI

Lemma2.1.AsasubvarietyofJ,thedivisorΘissingularonlyattheoriginofJ.

Aproofofthisfactisseen,forinstance,inLemma1.7.2(2)of[12].

Let(λ3x+2λ4x2+3λ5x3+4λ6x4+5λ7x5)dx(1)η=

2y

η(3),=x3dx

1

22 1,δ′=0

·n∈Z2 exp[2π2√uη′ω′2 1ttu)′′′′t′′′ 1t(n+δ)Z(n+δ)+(n+δ)(ωu+δ)}]′(2.1)

withaconstantc.Thisconstantcis xedbythefollowinglemma.

Lemma2.2.TheTaylorexpansionofσ(u)atu=(0,0,0)is,uptoamultiplica-tiveconstant,oftheform

σ(u)=u(1)u(3) u(2)

3λ7u(2) 42λ0332λ2u(1)u(2) λ2u(1)u(2) u(2)u(3) 3322λ3u(1)u(3)3λ66

45.

Lemma2.2isprovedinProposition2.1.1(3)of[12]bythesameargumentof[12],p.96.We xtheconstantcin(2.1)suchthattheexpansionisexactlyoftheformin2.2.