DETERMINANT EXPRESSIONS FOR HYPERELLIPTIC FUNCTIONS IN GENUS THREE

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

DETERMINANTEXPRESSIONSFOR

HYPERELLIPTICFUNCTIONSINGENUSTHREE

YOSHIHIROONISHI

1.Introduction

Letσ(u)and (u)betheusualfunctionsinthetheoryofellipticfunctions.Thefollowingtwoformulaewerefoundinthenineteenth-century.Firstoneis

( 1)n(n 1)/21!2!···n!σ(u0+u1+···+un) i<jσ(ui uj)

σ(u)n2

Althoughthisformulacanbeobtainedbyalimitingprocessfrom(1.1),itwasfoundbefore[7]bythepaperofKiepert[9].

Ifwesety(u)=1 ′ ′′= .. . (n 1) ′′ ′′′... (n)······...··· (n 1) (n)... (u). (2n 3) (1.2)

2y.

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

2 YOSHIHIROONISHI

Then(1.1)and(1.2)iseasilyrewittenas

σ(u0+u1+···+un) i<jσ(ui uj)

respectively.

Theauthorrecentlygaveageneralizationoftheformulae(1.3)and(1.4)tothecaseofgenustwoin[13].Ouraimistogiveaquitenaturalgenaralizationof(1.3)and(1.4)andtheresultsin[13]tothecaseofgenusthree(seeTheorem3.2andTheorem4.2).Ourgeneralizationofthefunctioninthelefthandsideof(1.4)isalongalinewhichappearedforacurveofgenustwointhepaper[8]ofD.Grant.AlthoughFay’sfamousformula,thatis(44)inp.33of[6],possiblyrelateswithourgeneralizations,noconnectionsareknown.

Nowwepreparetheminimalfundamentalstoexplainourresults.Letf(x)beamonicpolynomialofxofdegree7.Assumethatf(x)=0hasnomultipleroots.LetCbethehyperellipticcurvede nedbyy2=f(x).ThenCisofgenus3anditisrami edatin nity.Wedenoteby∞theuniquepointatin nity.LetC3bethecoordinatespaceofallvaluesoftheintegrals,withtheirinitialpoints∞,ofthe rstkindwithrespecttothebasisdx/2y,xdx/2y,x2dx/2yofthedi erentialsof rstkind.LetΛ C3bethelatticeoftheirperiods.SoC3/ΛistheJacobianvarietyofC.Wehaveanembeddingι:C →C3/Λde nedby Pdx Px2dxP→(∞2y,∞ x′ ′′ x= .. . (n 1)xσ(u)ny′y′′...y2(x2)′(x2)′′...(x)2(n 1)(yx)′(yx)′′...(yx)(n 1)(x3)′(x3)′′...(x)3(n 1)······...···(n 1) (u), (1.4)

2y,u(2)= (x(u),y(u))

∞xdx2y

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE3

withappropriatechoiceofapathoftheintegrals.Needlesstosay,wehave(x(0,0,0),y(0,0,0))=∞.

Fromourstandingpointofview,thefollowingthreefeaturesstandoutontheformulae(1.3)and(1.4).Firstly,thesequenceoffunctionsofuwhosevaluesatu=ujaredisplayedinthe(j+1)-throwofthedeterminantof(1.3)isasequenceofthemonomialsofx(u)andy(u)displayedaccordingtotheorderoftheirpolesatu=0.Secondly,whiletherighthandsidesof(1.3)and(1.4)arepolynomialsofx(u)andy(u),whereu=u0for(1.4),thelefthandsidesareexpressedintermsofthetafunctions,whosedomainisproperlytheuniversalcoveringspaceC(oftheJacobianvariety)oftheellipticcurve.Thirdly,theexpressionofthelefthandsideof(1.4)statesthefunctionofthetwosidesthemselvesof(1.4)ischaracterizedasahyperellipticfunctionsuchthatitszeroesareexactlythepointsdi erentfrom∞whosen-plicationisjustonthestandardthetadivisorintheJacobianofthecurve,andsuchthatitspoleisonlyat∞.Inthecaseoftheellipticcurveabove,thestandardthetadivisorisjustthepointatin nity.

Surprisinglyenough,thesethreefeaturesjustinventgoodgeneralizationsforhyperellipticcurves.Moreconcreatly,ourgeneralizationof(1.4)isobtainedbyreplacingthesequenceoftherighthandsidebythesequence

1,x(u),x2(u),x3(u),y(u),x4(u),yx(u),···,

whereu=(u(1),u(2),u(3))isonκ 1ι(C),ofthemonomialsofx(u)andy(u)dis-playedaccordingtotheorderoftheirpolesatu=(0,0,0)withreplacingthederivativeswithrespecttou∈Cbythosewithrespecttou(1)alongκ 1ι(C);andthelefthandsideof(1.4)by

1!2!···(n 1)!σ(nu)/σ2(u)n,

whereσ(u)=σ(u(1),u(2),u(3))isawell-tunedRiemannthetaseriesandσ2(u)=( σ/ u(2))(u).Therefore,thehyperellipticfunctionthatistherighthandsideofthegenerlarizationof(1.4)isnaturallyextendedtoafunctiononC3viathetafunctions.AlthoughtheextendedfunctiononC3isnolongerafunctionontheJacobian,itisexpressedsimplyintermsofthetafunctionsandistreatedreallysimilarwaytotheellipticfunctions.Themostdi cultproblemisto ndthelefthandsideoftheexpectedgeneralizationof(1.3).Theanswerisremarkablyprettyandis σ(u0+u1+···+un)i<jσ3(ui uj)2

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.

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

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.

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE7

Lemma2.3.Let beanelementofΛ.Thefunctionu→σ(u)onC3satis esthetranslationalformula

σ(u+ )=χ( )σ(u)expL(u+ , ),

whereχ( )=±1isindependentofu,L(u,v)isaformwhichisbilinearoverthe√real eldandC-linearwithrespecttothe rstvariableu,andL( 1, 2)is2π

u(j) u(k)logσ(u), jk···r(u)=

uσ(u),σjk···r(u)=(j)

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)

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE9

Lemma2.10.Letvbea xedpointinκ 1ι(C)di erentfromanypointofκ 1(O).Thenthefunction

u→σ3(u v)

vanishesoforder2atu=(0,0,0).Precisely,onehas

σ3(u v)=σ2(v)u(3)+(d (u(3))≥3).

Proof.Sinceu visonΘ,wehaveσ(u v)=0.Ifwewriteuas(x1,y1)andvas(x2,y2),2.4(1),2.5and2.7implythat2

x(u).Since

d(u(j) v(j))σ3(u v)σ(u v)2σ3=33 σ23σ x1x2+x2x3+x3x1∞= 1 x3= 1dxthederivativeofthefunctionu→u(j)onκ 1ι(C)bydu(j)du(j)dx

du(j)d(u(1) v(1))=dx

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

10 YOSHIHIROONISHI

vanishingordersofu→u(j) v(j)(j=1,2)areequaltoorlargerthanmby(2.3).Furthermoretheexpansion

σ(u v v1 v2)

=σ1( v1 v2)(u(1) v(1))+σ2( v1 v2)(u(2) v(2))+σ3( v1 v2)(u(3) v(3))

+(d (u(1) v(1),u(2) v(2),u(3) v(3))≥2)

showsthatthevanishingorderofu→σ(u v v1 v2)ishigherthanorequaltom.Hencemmustbe1.Ontheotherhand,2.2and(2.3)implythat

σ3(u v)=(u(1) v(1))+(d (u(1) v(1))≥2).

Thusthestatementfollows.

Lemma2.12.Ifuisapointofκ 1ι(C),then

σ3(2u)

33(2u) 22(u)2= 2σ33+3σ32 σ333σ2

σ22 σ22σ(u)

tothefunctionσ(2u)/σ(u)4,bringinguclosetoanypointofκ 1ι(C),weobtainthelefthandsideofthedesiredformula.Herewehaveusedthefactthatu→σ3(2u)doesnotvanish,whichfollowsfrom2.9.Thusthethefunctionσ3(2u)/σ2(u)4isafunctiononι(C),thatisσ3(2(u+ ))

σ2(u)4

foru∈κ 1(C).Lemma2.8(1)statesthisfunctionhasitsonlypoleatu=(0,0,0)moduloΛ.Lemma2.2and2.8(2)givethatitsLaurentexpansionatu=(0,0,0)is

2 13u(3)3

( u(3)3+···)4 (2u(3))+26λ7=2

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE11

Definition-Proposition2.13.Letnbeapositiveinteger.Ifu∈κ 1ι(C),thenσ(nu)ψn(u):=

σ2(u)σ2(v)22

Proof.IfweregardutobeavariableonC3,thefunction

u→σ(u+v)σ(u v) 1= 1 x(u) .x(v)

333

33

2(u v)

( u(3)3+···)2σ2(v)2= 1

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

12 YOSHIHIROONISHI

Theorem3.2.Letn≥2beaninteger.Assumethatu0,u1,···,unbelongtoκ 1ι(C).Then σ(u0+u1+···+un)i<jσ3(ui uj)

σ2(u)3σ2(u1)3σ2(u2)3

Proof.Wesupposethatu,u1,u2areanypointsnotonκ 1ι(C).Sincethesumof

pull-backsoftranslationsTuΘ+T u1Θ+T u2Θislinearlyequivalentto3Θ1+u2

bythetheoremofsquare([11],Coroll.4inp.59),thefunction

σ(u+u1+u2)σ(u u1)σ(u u2)σ(u1 u2) 1 = 1 1x(u)x(u1)x(u2) x(u) x2(u1) .x2(u2) 2

33(u u1)

22 333 222 33 22(u1 u2)(u)(u2)

tothefunctionabove,bybringingu,u1,andu2closetopointsonκ 1ι(C),wehavethelefthandsideoftheclaimedfurmula.Herewehaveusedthefactthatσ(u u1),σ(u u2),andσ(u1 u2)vanishforu,u1,andu2onκ 1ι(C)byLemma

2.4(2).Sothelefthandsideasafunctionofuonκ 1ι(C)isperiodicwithrespect

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE13

toΛ.Nowweregardthebothsidestobefunctionsofuonκ 1ι(C).Weseethelefthandsidehasitsonlypoleatu=(0,0,0)moduloΛby2.8(1),andhasitszeroesatu=±u1andu=±u2moduloΛby2.4(2),2.9.Theseallzeroesareoforder1by2.4(2)and2.11.ItsLaurentexpansionatu=(0,0,0)isgivenby2.8(2)and2.10asfollows:

σ3(u1+u2)σ2(u1)σ2(u2)σ3(u1 u2)

Therighthandsideis 1 1 1x(u1) x(u2) u(3)4+···).

Proof.Weknowthelefthandsideoftheclaimedformulais,asafunctionofu,aperiodicfunctionwithrespecttoΛ.Itspoleisonlyatu=(0,0,0)moduloΛandiscontributedonlybythefunctionsσ2(u)4,σ3(u u1),σ3(u u2),σ3(u u3).By2.8(2)and2.10,theorderofthepoleis4×3 3×2,thatis6.Thezeroesofthelefthandsideareatu= u1, u2,andu3moduloΛwhicharecomingfromσ(u+u1+u2+u3);andatu=u1,u2,u3whicharecomingfromσ(u u1),σ(u u2),σ(u u3),respectively.These6zeroesareoforder1by2.11.Thusweseethatthedivisorsoftwosidescoincide.Thecoe cientofleadingtermoftheLaurentexpansionofthelefthandsideis

σ(u1+u2+u2)σ2(u1)σ2(u2)σ2(u3) i<j 1 1= 1 1x(u)x(u1)x(u2)x(u3)x2(u)x2(u1)x2(u2)x2(u3)σ2(u)4σ2(u1)4σ2(u2)4σ2(u3)4 x3(u) x3(u1) .x3(u2) x3(u3)σ3(ui uj)

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

14 YOSHIHIROONISHI

ProofofTheorem3.2.Thebestwaytoexplainthegeneralstepoftheinductionisprobablytodemonstrateonlythecasen=4.Thecaseofn=4isclaimedasfollows.Assumethatu,u1,u2,u3,andu4belongtoι(C).Thenwewanttoprovetheequality σ(u+u1+u2+u3+u4)σ3(u u1)σ3(u u2)σ3(u u3)σ3(u u4)i<jσ3(ui uj)

u(j) v(j)=1

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

HYPERELLIPTICFUNCTIONSINGENUSTHREE15

Proof.Becauseof3.1wehave

x(u) x(v)

σ2(u)2σ2(v)2·σ3(u v)

u(j) v(j)

xj 1=dx(v)by(2.2).Theassertionfollowsfrom2.12.

SinceourproofofthefollwingTheoremobtainedbyquitesimilarargumentbyusing4.1asinthecaseofgenustwo(see[13]),weleavetheprooftothereader.

Theorem4.2.Letnbeanintegergreaterthan3.Letjbeanyoneof{1,2,3}.Assumethatubelongstoκ 1ι(C).Thenthefollowingformulaforthefunctionψn(u)of2.13holds:

(1!2!···(n 1)!)ψn(u)=x(j 1)n(n 1)/2(u)× (x2)′(x3)′y′(x4)′ x′

′′(x2)′′(x3)′′y′′(x4)′′ x ′′′(x2)′′′(x3)′′′y′′′(x4)′′′ x ..... ......... . (n 1)x(x2)(n 1)(x3)(n 1)y(n 1)(x4)(n 1) ··· ··· ··· (u)... . ···(yx)′(yx)′′(yx)′′′...

(yx)(n 1)(x5)′(x5)′′(x5)′′′...(x5)(n 1)

Herethesizeofthematrixisn 1byn 1.Thesymbols′,′′,···, 2 dd,···,du(j)(n 1)denote

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

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FacultyofHumanitiesandSocialSciences

IwateUniversity

Morioka

020-8550

Japan

onishi@iwate-u.ac.jp

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