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

16 YOSHIHIROONISHI

References

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8.D.Grant,‘AgeneralizationofaformulaofEisenstein’,Proc.LondonMath.Soc.62(1991),121–132.

9.L.Kiepert,‘WirklicheAusf¨uhrungderganzzahligenMultiplikationderelliptichenFunktio-nen’,J.reineangew.Math.76(1873),21–33.

10.S.MatsutaniandY.Onishi,‘Determinantexpressionforhyperellipticfunctions’,withAp-

pendixbyS.Matsutani,Preprint,(2001),/abs/math.NT/0105189.

11.D.Mumford,Abelianvarieties,(OxfordUniv.Press,1985).

12.Y.Onishi,‘Complexmultiplicationformulaeforhyperellipticcurvesofgenusthree’,(thecorrectionisavailablefromhttp://jinsha.iwate-u.ac.jp/kankyou/onishi/sup.ps),TokyoJ.Math.21(1998),381–431.

13.Y.Onishi,‘DeterminantexpressionsforAbelianfunctionsingenustwo’,Preprint,(2000),/abs/math.NT/0105188.

FacultyofHumanitiesandSocialSciences

IwateUniversity

Morioka

020-8550

Japan

onishi@iwate-u.ac.jp