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

HYPERELLIPTICFUNCTIONSINGENUSTHREE5

2.TheSigmaFunctioninGenusThree

InthisSectionwesummarizethefundamentalfactsusedinSections3and4.Detailedtreatmentofthesefactsaregivenin[1],[2]and[3](seealsoSection1of

[12]).

Let

f(x)=λ0+λ1x+λ2x2+λ3x3+λ4x4+λ5x5+λ6x6+λ7x7,

whereλ1,...,λ7are xedcomplexnumbers.Assumethattherootsoff(x)=0aredi erentfromeachother.LetCbeasmoothprojectivemodelofthehyperellipticcurvede nedbyy2=f(x).ThenthegenusofCisg.Wedenoteby∞theuniquepointatin nity.Inthispaperwesupposethatλ7=1.Thesetofforms

ω(1)=dx

2y,ω(3)=x2dx