DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS(4)

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

4 YOSHIHIROONISHI

processfrom(1.3),ourgeneralizationof(1.4)isobtainedbysimilarlimittingprocessfromthegeneralizationof(1.3).

Althoughthispaperisalmostlybasedon[13],severalcriticalfactsareappearedincomparisonwith[13].Sections3and4aredevotedtogeneralize(1.3)and(1.4),respectively.WerecallinSection2thenecessaryfactsforSections3and4.

TheauthorstartedthisworkbysuggestionofS.Matsutaniconcerningthepaper

[13].Afterhavingworkedoutthispaper,theauthortriedtogeneralizetheformula(1.3)furthertothecaseofgenuslargerthanthreeanddidnotsucceed.Theauthorhopesthatpublicationofthispaperwouldcontributetogeneralizeourformulaoftype(1.3)or(1.1)tocasesofgenuslargerthanthreeinthelineofourinvestigation.Matsutanialsopointedoutthat(1.4)canbegeneralizedtoallofhyperellipticcurves.Thereaderwhoisinterestedinthegenaralizationof(1.4)shouldbeconsultwiththepaper[10].

Cantor[5]gaveanotherdetermiantexpressionofthefunctionthatischaracter-izedinthethirdfeatureaboveforanyhyperellipticcurve.TheexpressionofCantorshouldbeseenasageneralizationofaformuladuetoBrioschi(see[4],p.770, .3).TheAppendixof[10]writtenbyMatsutanirevealstheconnectionofourformula,thatisTheorem4.2below,andthedeterminantexpressionof[5].Sowehavethreedi erentproofsforthegeneralizationof(1.4)inthecaseofgenusthreeorbelow.

Therearealsovariousgeneralizationsof(1.1)(or(1.3))inthecaseofgenustwodi erentfromourline.Ifthereaderisinterestedinthem,heshouldbereferedtoIntroductionof[13].

Weusethefollowingnotationsthroughouttherestofthepaper.Wedenote,asusual,byZandCtheringofrationalintegersandthe eldofcomplexnumbers,respectively.InanexpressionoftheLaurentexpansionofafunction,thesymbol(d (z1,z2,···,zm)≥n)standsforthetermsoftotaldegreeatleastnwithrespecttothegivenvariablesz1,z2,···,zm.Whenthevariableortheleasttotaldegreeareclearfromthecontext,wesimplydenotethemby(d ≥n)orthedots“···”.Forcrossreferencesinthispaper,weindicateaformulaas(1.2),andeachofLemmas,Propositions,TheoremsandRemarksalsoas1.2.