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(1)RIEMANN'S PERIOD MATRIX OF Y2=X2n+2-1 Seiji YAMAzAKI", Minoru ITo, Yoshimitu SAITO,. ToshikazuIKEDA"'. MasahikoKuBo,"' MasaakiTAsHIRo"'", TeiichiHIGucHI" May 1, 1995 Abstract This paper introduces a rule to get Riemann's period matrix of y2=x2n+2-1. By this rule we can get unique result which satisfy the Riemann's period relation for a homology of Riemann's surface. In fact, we show two matrix :( n,T ') and (' T ,' rr ') for g2=x2n+2-1. Still more, we look for the modulai matrix:7i=z-1n'," 71="n-1"n'and the modular translate. SE SL(2n,Z ). The reader can ascertain that Tand "T are symmetric matrix in n =2,3.. "Yamanashi University "" Yokohama. National University. " " Kyourin University. " "Tokyo University of Agriculture and Technology.

(2) 12. Seiji YAMAzAKI, Minoru ITo, Yoshimitu SAITo, Toshikazu IKEDA,Masahiko KuBo, Masaaki TAsHIRo, Teiichi HIGucHI. 1 Introduction We read Tata Lectures on Theta 1 which written by David Mumford and was interested in Reimann's period matrix in chapter 2. Therefore, we are going to make example of Riemann's period matrix for the very particular case of hyperelliptic Riemann surfaces.. The first step is to choose a base of the vector space of holomorphic 1-form on Riemann. surface X for y2=x2n+2-1. The second steP is to make a model of Riemann surface for y2=2n+2-1.As a topological space, X is a compact orientable 2- maniforld. Its genus is n and its branch points are (2n+2)th root of 1. The dimension of the space is n by Riemann-Roch theorem.. The third step is to analyze the periods of the holomorphic 1-form by using Green's theorem. To do this we have to cut X on a base of homology- 2n disjoint simple closed paths, all beginning and ending at the same base point.. The periods have to be decided so that can satisfy the Riemann's period relation. And. therefore, we determined five rules to get correct the periods from a polygon with 4n. sides (one side each for the left and right sides of each path) Using this rule and Cauchy's integral theorem, we succeeded in standing for the value of each periods by. ' period value of the branch point. The thema of this article is such a way to get Riemanns matrix of y2=x2n+2 - 1. In the latter half of this article, we look for Riemann's period matrix for a different base. of homology and try to form two Riemann's period matrix by modular traslate SL(2n, Z).. 2 General Theory of Hyper-Elliptic Function 2.1 Period Matrix and Quasi-Period Matrix We think a following hyper-elliptic curve C.. C:g2=Ao+ A ix+ A 2x2+'" + A 2n+2x2""2 A,EC (i = O,1,･'',2n+2) (1)Genus of hyperelliptic curve C is n. And therefore, the vector space of holomorphic 1-. forms is n'vector space by Riemann-Roch theorem. In fact the following is a concrete. base.. cvi=!dx, w2=e[Idx, w3=-X2du,...,wn.xn-i y. yyy. (2)The second kind of differential form which has pole on oo is n -vector space. In fact,. the following is concrete base.. 2n+2-i. 77i=y-1 ;.i (k+1-i)Ak+i+ixkcix (i=1,･･･,n).

(3) RIEMANN'S PERIOD MATRIX OF-. Y 2. x2n+2bl. 13. (3)Homology group on Riemann's surface for hyperelliptic curve C has 2n generator Ai, Bi. (i =1,"',n )which satisfy a following condition.. 1) Ai×,t!i= Bi×Be=O(i =#j). 2)AiX&=aii･ (i,j=1,･･･,n) We define Riemann's period matrix n = (fi,ll ') from (1),(2),(3).. Definision 2.1 (Riemann's period matrix) For a base ofholomorphic 1:fornz tui(i =1,..., n ). and a base ofhomology A i,Bi(i =1,..., n ) on Riemann's sudece. Riemann's period matrix St and quasi period matrix fl are delfinedfollowing.. periodmatrix:n=(ll,n'). ll n-C': :il ;i.n.) n-C･,:1 'i'.;'i.n.). T,]=fCOI T',7=fCUi. AJ' Bj Quasiperiodmatrix: ft'=(rt,rt'). fi-( /i･i,i:Il.:il) fi'-( ･l:llll/ril:). fi;,-f7i fi,',-fqi. AJ' Aj ' Lemma 2.1 (Riemanns period relation 1) Let X be a compact Riemann suJface ofgenus n,. withcanonicaldissectionX=XoUAiU'"UBiU"'UBn･ For any holomorphic 1-form to i, wj (i ij' =1,...,n ),the period satisfy a following equation.. ,2n.,[ f .,Òi f .,`C' J' -f .,`Vi f .,Ò,j ]=O. Lemma 2.2 (Riemann's period relation 2) Let X be a compact Riemann surface of genus n,. with canonical dissection IY±XoUAiU"'UA.UBiU'''UB. .For any holomorphic 1form wi (i=1,...,n ).the period satisfy a following equation.

(4) 14 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. i,E.,[f .,cvif .,dij･ -f .,coif .t-･] > o. namely,. ihi i"ll.,(f.,(V- f .,Ò )>O. Theorem 2.1 (Riemann) For the Riemann's period matrix fl=(n,ll') of hyper-elliptic. curve C. . ModularmatzixTLn'in':symmeuicmauix Proof.. Bylemma2.1 foralli,j(i,J'-1,...n). ' k2.1 [ rr. nik. n 'Jk- rr tik rr yk ]=o. '. (nti rrtn)C.I':)-(nii'"Tin)C.Ij:)-O. ( Ttl '" Ttn )t( n 'jl '" nYn )-( rr h '" n'tn )t( Zji "' Tjn)=O. C'::Ii･rrnl.".)t( /'':1.'l'il//"l#.)-( :'1:1'ii･//":.)t( /･::1'IiZEi#.)"o. Htn'=H tt n (*) Here, Iet ' toi"',' to.be a other base of holomorphic 1-form which differ from toi"', w. such that. ' ' (Itual)""(lca5;)=(liii,'li.AA･il.)(Z'1). then period matrix " S) =(H,ll ') for a base " w i'",' to n is following.

(5) RIEMANN's pERIoD MATRIx oF y2=x2n+2-1 ls. 'Ti, .41, 'coi-.(l, A. Aiicoi- ;.l, Aii.,4L, òi A., Aiinb. 'TtJ f,, 'wi-f,, 22., Aiicoi- il.il, Aii.1[, (vi A., Aiin'b･. ')rij=.(A,ii'"･Lin)(i.:.) "-'ii=(･tii'"?Lin)(ll.Il'),). ' (Il'il'ii.',;'l.)"( lil'ji.A,'L)( :;ilTi//:l): '"="". ( li'l.'lTi'*T,,ll.". )=( llil'ii.AA':1 )( /1.'l'il;'li.): ""'=""'. ' 'S')= (' H,' n')=(A n, A nT) = A(n,H)= A SIL. If A=H"'(detntO) * S-) = (n ," n') = ( ll, H " H') = (IL,,T). Using equation ( " ) in lemma 2.2 for ' n. * I]I t(* lll) =* fi lt(* " ). By'H=lh:(unitmatrix)and"H'=T Theorem 2.1 means that there is a suitable base of holomorphic 1-form so that A-period matrix: fi can be a unit matrix za, B-period matrix: H'can be a symmetric matrix Z Thus, T=ll 'i n is a symmetric matrix.. Theorem 2.2 (Riemann) For the Riemann's period matrix n =(n,H ') of h>per-elliptic curve. C im T FIm( H -' H'):Real symmetric matrix ofpositive definit sign Proof.. -iKEJtn=-i(n-HTi-r)(-Oih ioh)(iiHil)=i(t,tn-n-tii,).

(6) L. 16 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. From a above equation, -isiJtfl is Hermite matrix. Making a following Hermite form for this matrix,. -- -- -- nnn nnn. A(-i()Jtfl)tR=i(Ai'''An)(H'tII-II'II')t(Rn"'Rn). =i[2 2 2(Ai7rikjljfiy･k) -i[2 2 2(AiiT･'ikjZjjijk)]. k=1 i=l j--1 k=1 i=l j=1. nn nnn. =i ill.i, [L,,( i.Illl.,Aicai).Jlb,( i.,Aitoi)-./),( ;..,Aitoi) .L,,( ;..,Aiwi)]. nn. Let 'wi "dii be (2i.i Aitoi), (2i.i Aiwi) by Iemma 2.2.. nn n n. '. 7( -iSi ii Jtn )t 1-i =i il.l,.[.L,,( i.Iil. ,Ai coi)Jlb,( i.El. ,Ai tui) - J),( i.Il. ,Ai coi). Likewise theorem 2.1, if we choose a suitable base of holomorphic 1-form, period matrix: fl =( fi,fi') can be ( In,T). Thus using the above mentions for ( lh,T). -i(inT)(-Oih ioh)(,iTh)=i(T-T)=2imT>o. which proves lemma 2.2 2.2 Hyper-elliptic-function of C. In this section, we will represent Abel-function on Jacobian variety by theta function of several variables. Theta function of several variables is given by a following.. '. @(7,T)=2exp[ni`n-Tn',+2zi'n-z-] . nEzn. '. @( Z,T)is entire function which has a quasi period-A = IZ" + TZn.Iis a unit matrix and Tis a symmetric matrix.. e(7+ nt,T)=0(Z,T) e(Z+ nt,T)= exp[Titn-' Tn',+2Ti'n-z-]e(Z,T) For this @( Z,T), a entire function a(2) is defined by a following. o(z)=exp[-5z-rin-'tz-]e Ss': (n-itz-',T).

(7) RIEMANN's pERIoD MATRIx oF y2=x2n+2-1 17 however, s". t( -21 ,..., -21 ) o ,, . t( :, n2-1, ..., -21 ). - (Z,T) =-2 exp[Ti'(n- + d)T(n- +d)+2ni'(n' +a-)(z- +b-)] @ b"nGzn z----(z,,...,z.) z,=f(Xi'Yi)to,+...+f(Xn,Y")to,(,.1,...,n). oc oc Definision 2.2 (Hyper-elliptic-function) For i,jE( 1,...,n ). 02. piJ'(2)= - o.io.i iog O'(7). are Abel-:function on the lacobian varietly ofhyper-elliptic curve C. piJ'(Z)satidy a following eguation. Particularly, when genus nisl. p n (z) is a Weierstrass' p :fiznction.. ￡., tl., pij(z)xP''xl'i F(Xii,X-sl-,2)y,rys. 2 npij(z)x;" =o (r,s=1,...n) i=1 However; F(.Xi,Xli)=A2n+iX? X: (Xi,+.Xi])+ n-1 2 Xl XS (2A,,+A,,+1(.Xl,+.Xli)l. i--o. 3 PeriodMatrixof y2=x2n'2-1 3･1 The Way of Deciding Period Matrix of y2 =x2"'2 -1. ' 3.1.1 STEP 1: A Base of holomorphic form 1-for yp =xq-1. 'Xb. 7S?lql g13Jl Let N iS dimension ofvector space ofholomorphic 1:form giT. drfor gp =xq -1. ' N.(P-1)(q-1)･ 2.

(8) 18 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. Proof. (a)(x,y )=( ai, O) (i = 1,...,q). ai is q-th roof of 1. Bypy P-i dy=qx q-i cix, Order ofcix's zero point in y = O is p-ls. ' Xb y-. (ix is holomorphic in (a)ol <- a <- p-1 (b)(x,y)=(oo,oo) As we can view y p =xq -1 as y p =xq in (b), we can put x, y, on t-p,t 'q By (ix= -pt-p-qdt,. p. xb t-op. g-. du= F., (-Pt-P-i)dt = -ptaq -bpmip+i) dt. Xb y-. (ixis holomorphic in(b)o aq-bp -(p+1) ZO. N - .il.il, [aq-p(p+i) ]+i. = -21 .#1["q-SP'1) ]+1+ -i .gl[ tp-a)p'tp+1) ].1. ' =･ -2i .E.ii (M-i)'i"-i ￡.i (q-m-2)'i aq=mp+r (o <--r<- -i). = -2i .i.lii (q-i)= -2i (p-i)(q-i). '. 3.1.2 STEP2: RiemannSurfaceof y2=x2n+2-1 At first, let think about Riemann Surface of y2=x. i.e, y =V3i .We put the value of g,xi=re è. x2=ret(e'2n)on y,, g,. As we can get y,='y, by easy calculation, y=a is one-to'two mapping from x"surface to y -surface x i.e. 2-valued-function which is showed by a below. figure. ,.

(9) RIEMANN'S pERIoD MATRIx oF y2=x2n+2-1. 1. ,. o. 19. X2. Y. ,Y, o. o. o. ,. But y= V3E is not2-valued-function in x=O which satisfy y= v5i =0. And as we can get. t =ls again by putting x, y on 1/s, 1/t.y=V5i is not 2-valued-function in x= oo too. Thus x= O and x = oo are branch points of R,. R,is the thing which joined up-side of x,' surface to under-side of x,-surface and under-side of x,'surface to up-side of x2-surface. formally by cutting two surface x, x, along segument connecting two branch points. x=o,oo. Next let think about Riemann Surface Rn of y2= x2n'2-1 like g = v(I. Rn has two sheets. yi= x2"'2-1 and y2= ndyi= x2""2-1.As we can get t2=s2"'2(1-s2""2)-iby putting x,g on 1/s, 1/t in g2=x2""2-1, branch points of Rn are (2n+2)`h'root ofx=pi. (i=1,...2n+2). To see the model of Rn, we cut two sheets foll'owing so that the sheet can change by rounding each branch points one time. P2 .. ,. R. .. '. ====nt. '. Bn+2. ag+1. i. 2 li. wBn ""'" %"d Bn+2. L---... RB rg e?.--fg. u P2 Bn+i. let join xi-surface to x2-surface like g = VI.. This is Riemann surface Rn. of y2=x 2""2-1. It genus g is dimension of vector space of. holomorphic 1' form by Riemann-Roch theorem. Therefore, lemma 3.1.1 g = 2((2n+1l/2=n.. t. RB. Be. PLp. EB. 1 ii.+, B.., RWP g';'::1[lli' >tg""'nj-2. 2. J ); l,liii..-.:.g;."i"'" Bn+2.

(10) 20. Seiji YAMAzAKI, Minoru ITo, Yoshimitu SAITo, Toshikazu IKImA,Masahiko KuBo, Masaaki TAsHIRo, Teiichi HIGucHI. 3.1.3 STEP 3: Five Rules ofDeciding Period Matrix We decide a base homology for Rn which is made in Step 2. Following figu･re show it.. Ai A2 An. AiXBi=1 (i=1,".n) . : : 1 ' '- . A,xAj--･B,xlib=o (i=#j) ･･･B, lliliilll:Eiiiir2 Here, let put cross point of Ai, Bi on Ti (i =,...,n) and we get together 71 to branch. point oo from same direction.. Ai A2 An. '･, ' ･ 1 "x--z71E'. , ------d -,.- -- ------ -----B, 'SB2:Sl.2-.---bli.;EBI!. Next, we cut and open Rn along the base of homology Ai, Bi. It becomes a sheet of simple connected domain namely, 4n'polynomial which has 4g-side At･,A7･ ,Bt･ ,B7･ (i"1,...,n )provided that At･, Bt･ ,shows right side of Ai, Bi and A7･ ,B7･ shows left side. Ai, Bi. At last, we have to write difference of two sheet of complex surface xi, x2 and arrangement of branch points in 4n-polynomial.. On the above mentions, we decide the periods of holomorphic 1-form for Ai, Bi by. making some simple ,a{, '`"", "i ,tot,ai' i<illlli.lill, ,,, "ii,ii]ii<lx<pe?'`,,,s, i... 'AN. n" 1. i3il`2. le. 3. M4 2 nn sei n4. E")-1. 2. e. ･2 '. 2, (N)"-. "?". eF. .2 M4. B5M3' r!4 n3. 2. 2,. ' ptsBpt+3Q. s. ab. pt5. 2. f"6 B'3 ""6. "6 M. M closed path which pass branch points. At' this time, Cauchys' integral theorem play the leading role. But the periods of holomorphic 1'form for Ai, Bi must be decided uniquely so that may satisfy Riemanns' period relations i.e. th eorem 2.1 theorem 2.2. Therefore, to realize this object, we state following five rules..

(11) RIEMANN'S PERIoD MATRIx oF y2=x2n+2-1. 21. e Rule1 : A simple closed path must include right side AI,Bi. e Rule2 : A simple closed path must include even branch point. e Rule 3: The sign of holomorphic 1'form in the path which get out from starting points. of Ai, Bi must be same sign in the path which get into end points of Ai, Bi e Rule 4: The sign of holomorphic 1'form in the path which get out from a branch point must be different from sign in the path which get into same branch point.. e Rule 5: The sign of holomorphic 1-form in the path which connect two branch points. must be unchangeable. Remark 1: As the side of 4n-polynomial, -Bi .are enclosed by A+ i and Al･ .We can look. on the path of Bi as the path which connect a point mi, on At･ and Bti by Cauchy's integral theorem. Thus we decide the periods of Biforthis new paths by using above five rules.. Remark 2: As the side of 4n-polynomial: At are enclosed by Bt and Bj- (i ± j ) We cannot choose a common point from the points on Bt and ll-i.But at this case, we can decide value of period by making simple closed path which some pair of the side of 4n-. polynomial:At･,Al･,Bt,BT･. By using above five rule and two remark, we decided periods of holomorphic 1-form. A base of holomorphic 1-form for y2= x 2""2-1 is following by lemma 3.1.1.. toi=. - y cix bcix･ to2"g(ix･ ca3= x-g-dx,-,ev.= x2 ･xn-i. Namely,. coi =. xi-1 xi-1 y. cix= cix(i=1.",n) x2n+2-1. And let stand for cat on xi'surface, wt on x2"surface by caii, toi2･. X i-1. -Xt-1. Wtl=cat= x2n+2- 1du,tot2=-tot= x2n+2-1du.

(12) 22. Seiji YAMAzAKI, Minoru ITo, Yoshimitu SAITo, Toshikazu IKEDA,Masahiko KuBo, Masaaki TAsHIRo, Teiichi HIGucHI. 3.2 Period matrix of y =x2n'2-1 3.2.1 Calculation. 1. A-PERIOD MATRIX (1)Forj =1,2,...,n -1 By remark 2, we make a simple closed paths which start from a. in start point of A;, pass the sides At,i Bf.i A7.i B].i, At.2 Bt･.2 A;.2 B;･.2,..., A+. Btl A; B-.,arrive. at a. in start point of B. and start from its R., pass the branch points ,R,l>,..., Ljq,. Pbj , come back B. in end point of Aj . As the sum of integrate value onsides of 4n polynomial is O at this time, integrate path of Ajis following.. '. Rn+2;RSL-;L-tLL ･･･ Bj'iiBj.- An+2 (2) Forj --- n. We make closed simple path which include A. and pass the branch points Pl,B, fk, a, "' ,. a.-i, l)>.. Thus integrate path of A. is following.. an+2;RSR;L-±LR ･･･ Bn-iSAn;Ln-2 We get period rriJ' by above integrate path. 2. B-PERIOD MATRIX (1)Forj --1, 2, "', n-1 By remark 1, we can view the path of Bi as the newpath which. connect the point m;, on At and the point m;, on A;.Therefore, we make a simple closed paths which enclosed the new path and pass the branchpoints R]j, R2j.i. At this time, integrate path of Bjis following.. mL .+ IZ,., ; R. -# mS (2) Forj=n We make closed simple path which include B. and pass the branch points R,R, A, R, "' ,Ri.,. a.+i･ Thus integrate path of & is following.. an+2 S -PIin+i ; Ln ; Ln+2.

(13) RIEMANN's pERIoD MATRIx oF y2=x2n+2-1 23 We get period rdj by above integrate path. In next page, we show the period matrix H andH'concretely. At that time, you find that. matrix of n and H are matrix of vandermonde.. 3.2.2 Result. 1.PERIOD MATRIX A'period matrix fi: By above calculation, niJ: -2 Ef., (p6,m, - P6,)Kl (i,j-1,2,･･･,n). H.2(KiKi, O )(lll:i,t'Ii.lil3 pPl,:PajP,plil:R.p;?111 27-i(Ril-R,i,). 27.,(p2 -p2)). No KhApc-p!pr-ps+p?-p2･･･27.,(PIIt-i-P;t)/ B-period matrix H ':. By above calculation, n' iJ･=2(pSj",-p6j.,)Kl (i,j=1,2,･･･,n). ' (iiK> 2)(Iii,l7-ER:?, ft,T-ii.iiiz, .k21nl-:'/i#"Ji;). ".2. 2. DETERMINANT det H= 2KiK>･･･Kh (P - P2)(P2 - p4)･･･(pn- p2n)H=2(- 1)n.Kp "("")i2""') c. det ll'= 2KiK>･･･Kh (P2 - P3)(P4- ps)･･･(p2n- p3n)H=2(- 1)nKp n(n'i2("'2). '. Let put Ck on k" (P2i-1), then H is following vandermonde determinant.. i=1. H.. 1 P2 P4 .･. P2n-2 1 P4 P8 ... P4n-4 .................................... -------------e{-e--t----------------. 1 P2nP4n." p2n2-2n. =pn(n+1) i2n"1)chH1G-2G,-3･･.QCI.

(14) 24 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. n K=KiK･･･K. Ki= Jloo wtC=CiC2"'q-iCZ C"-' ,H.,(Pi-1) pt.--pi (i---1,2,･･･,n) P=cos(-:.li-, )+i sin ('{l.i-i ). 3.2.3 Confirmation for lemma In this section we show that the value of period made by five rule satisfy Riemann's period relation 1.2.. n nk. iE,) [.L,,wiJL,toJ]= IE,1 [I2 iE,) (p{t-i-pi,)Kll( 2(ps,-ps,.,)Kl11. .4KK, (1-Pt)(,1-Pj) E.), [ps, II.ii, pli]. E,) [11,wt.J],,caj]- #, [./l,,wtJl,,toJ]-O. which satisfy Riemann's period relation 1.. i il.ill, [.IL,wt /L, diJ] - L, tot.I),, dij]>o. n [!),, totL, diJ] - L, toiL, diJ]>o i ii:. 1,. im iE.i, n(IL,w-j ./ll,,wt)>o. On the other hand, we have only to prove last inequality for Riemanns ,period relation 2.. n nk. ;., (/L,diill,,toi]= iE.), [I2 ;.i, (PSi-iPLt)KllI2(P>,-(p>,.,)K,ll. 4Ke(lmPi)In+(n+1)P`l l1+pt12.

(15) RIEMANN'S pERIoD MATRIx oF y2=x2n+2-1. 25. im il.ii, l ./l,,ca-i.IL,wi]=1i4.Kpll,(2n+2)sin( .T.i)>o. 1. Above mentions show that Period matrix calculated for a homology in five rule satisfy Riemann's period relatioh. At the same timem we can get the value of integral for all 1-. cycle on Riemann surface of y2= x2"'im1. Namely, let z be any points on Rn then for a fixed point zo on Rn, we can solve the following problem.. X 7.;;l3ii,-f,ZO 7.fki,+2. Still more, we can find a relation that is lead from two period matrix for different base of. homology. We will state about this relation later. At any late, thereader will sure that a. simple closed path and period for each base of homology -Ai Bi (i = 1,"･,n)-is decided by five rule uniquely.. 4 Hyper-Elliptic-Function of y2=x2n+2-1 4.1 Hyper-Elliptic-Function of y2= x2n'2-1. In this section, we seek quisi-period matrix and define the hyper-elliptic-function of. g2=x2n'2-1 again.. A-QUISI PERIOD MATRIX fit: By above calculation, fi 'iJ' .2 2J,' .,(g?-",-(i-')) - 1tin-(t-i))Ki (i,j=1,2,...,n). Rim-R2n R2n-R,im+Eim-R,2n ･･･2.,(P&E,-P;,"). ---. fi=2(k) Rn+2-R,n+2 Rn+2-Il,n+2+Bn+2-Ln+2 ･･･ 2,n.,(4,Z;2-Rb:'2) Rn+i-,Ft,n+i Rn+i-R,n+i+Rn+i-.R,n+i ･･･ 2.,(.R,7-",J-P,:'i). B-QUISI-PERIODMATRIX H,: By above calculation, fi 'iJ' .2(I]7-"i"-i)- efn'(i-i))Kl (i,j.1,2,...,n). B2n-nj R2n-Rs2n･･･ llSn-RS.,. r". ---. t -v - - -. fi=2(K) Bn+2-Rlt+2 Rn+2-1t}t+2 ･･･ Bn."2-R￡l2, Bn+1-RIz+i Rn+1-1ti+j･･･I}:'i-,R;2li.

(16) 26 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI i. '. Hyper-elliptic-function y2=x2n'2-1. '. ' ' exp [--ll-z fiii "ti]@[ toij(z) =- o.l.]is., iogo (z) a (z)= :i I' ](ll-'ti,T)(i,j --. i,...,n) '. sn.t(!,...,-!) a.t(a, "-i ,...,!). 22 222 ￡., te.,pij(7)x,h-i xg-i= F(Xi.'l,S-)-.2,)Y,"YS i., pin(z)x;-i=o (r,s.i,...,n). However; R Xi, .X>)= Xni, Xn,(.)fi+X>)-2. 4.2 Concrete example n=3 4.2.1 Period and quisi-peribd matrix of y2= x8-1. Let think about Riemann Surface Rn of y2= x8-1 like R3 has two sheets yi=V Eg:T andy2= -yi - V-iig:T Branch points of R3 are 8`h-root of 1xFpi (i=1,...,8). To see. ･. the model of Rn ,we cut two sheets following so that the sheet can change by rounding each branch points one time. P2 P3 Pl. ･- 2 1. R. ----v--p-p-n 9Be9 rg fi} eB. Ps Ps P7. P6. let join xi' surface to x2' surface like y = V3i.. This is Riemann surface R3 of y2= x8-1. It genus g is dimension of vector space of holomorphic 1-form by Riemann-Roch theorem. Therefore, by lemma 3.1.1 g =(2 - 1). (7-1)/2=3. 1. R. ra B ee. se. Fili. 2. 1. R. '"'" lj. B"""ti ?""'B ii"2- B.

(17) RIEMANN's pERIoD MATRIx oF y2=x2n+2-1 27 ,. We decide a base of homology for R3 which is made in Step 2. Following figure show it. t tt. ,A2. Al A, A3･ e.. 's-' -' t. k･-`･Bliii,illl･････B,,,,,'N,･-･Niiil･- ,A,,...B'B4191iiA3. Here, let put cross point of At Bi on 71 (i=1,2,3) and we get together 71 to branch point. P2n+2fromsamedirection. .. Al A2 A3･. '. et t't t tt-. ect. l: Blx B2':B3's. '. F?.... .B rg'- ･'R l." "･--･ ag. B2. A2. --t .･･.. BR ,. m. e. m. '. '. .. 2.. - --. R rg. va. N Ms N. N. N. N B .:. t. R. 1. N. '. nl. Al, ss. fe. t. '. t. Bl. ett. e. tt. tt A3･. E}3.

(18) 28 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucm. '. '. Next, we cut and open Rn along the base of homology Ai, Bi .It becomes a sheet of simple connected domain namely, 4n-polynomial which has 4g-side At･ ,Al B:,BI･ (i =. 1,2,3) provided that At B:,shows right side of Ai, Bi and A:･,B;･ shows left side Ai, Bi. At last, we have to write differece of two sheet of complex surface xi, x2 and arrangement. of branch points in 4n'polynomial.. ptlilpts A'3. "s2 1'pts65. x. "7, ' e '. 21R e'. B; f"2 B. p(5. pt1 -. . 21.. N"fti'' 2･ rg1 e R, , 2 ivs-EiE '. M3. "3. E;rr!4 s2 n3 n4 3 ms m3", n4 Igii. .-A$. B5. et A3. A'3 Alti'2 ･1 e B5 1 Ei'I'. af. 2. 2R ,eg3 il) e!x.. B; B, l A'. e. 1. ･2 'k,,. rg. ,. 1. BT ttt: ptt --l. B 2. B. A'. i. Bl 2. B:. 4 :'. B'2. R. lg 2 " 2t A'2. 1. 2 2. A:. c. '"'. ,tN;. R･ -!d----... B'2. aj.

(19) RIEMANN'S pERIoD MATRIx oF y2=x2n+2-1. 29. A - PERIOD MATRIX n: H.2(.)(PA',:R.f .P,i,:AR?:AR?zRR? ;i,,zAp+.A,?:ftR?+.kR-?mi,kl?. XP?-Pg PP-PS+P8-P2 P?-PS+P8-P2+Pg-P8. P- P2+P3+1. n-2(K)( Pp,,-'. IP2. o. N p3+ p2. p3+ P2 - P+1. ( :p,;.2-p-,iP3. P3 - P3 P2+ 1. P+1. ). ). -.-p+.P,3 .P,3 Li. >(.)-,. H-i.2. P3-P P3+1 / B-PERIOD MATRIX Ht:. ) XP8-Pl M-P; P8-rv. H,.2(K)(lillii:Prk Ri,'ii,'-P,fu ill):Pi,g. H,. 2(K)( ;i: IP3 e: pl, Pp3,:p12. X-P- P2 P3-1 P- p2. ). MODULAI MATRIX T:. ) lnlX-1+p2-p3 p3 1-p2-p3. T.-.2-(i- Pp2,- P3i - pe3-- p,- i+p 2i, p3. Thus. (. 1+ V2. V2(1 - i). T- 1rrl. -1. 1- M. -1. 1- V2. 1+ V2. -1. -1. 1+ ta. ).

(20) 30 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA.MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. 5 Modular translation 5.1 General theory of Modular translation. When we seek Riemann period matrix of y2= x2"'2-1 , we chose a base of homology like figure. In this section we search a Riemann's period matrix for the other base of. homology and state a relation for two period matrix-(H,n') and('ll,' n). Here, if we take other base of homology :l"Ai, "','An,"Bi, "'"Bn l which stand by liner conbination. of one base of homology:lAi, "', An,Bi,"'Bn l for Rn for y2= x2"'2-1 then,. *Al i. an"'ain bii"'bin. 'An *Bl. a2i'"ann b2i"'b2n Cll"'Cin dii'"dln. i *Ein. i Ill I III. C2i"'Cnn d2i"'d2n. Ai I. An Bi l. th. '. A=(￡iiilllZ'.i) B=(//':illlZini). c-(g･1,'111C.lk･:) D-(gil,'111d,li). s. ({lg) We remark that a base of homology :I Ai,'"An, Bi,'"Bn lis standed by other base of homology I 'Ai,"'"An, "Bi,"''Bnl too. Thus we get a following. det S=1. AB1-1 CD. As i 'A i,'･'"An, "Bi,""Bnl is a base of homology for Rn, 1) *A, × 'Aj 'B, × "Bj =O (i =i Li). 2)"A,×"Bjl=Oij (i,j=1,-･,n).

(21) RIEMANN'S PERIOD MATRIX OF y2.x2n+2-1. 31. As (Ai,"'An, Bi,"'Bnl is a base of homology for Rn too. 3) A, × Aj =B, × Bj =O (i =l Li). 4)A,XBj=6ij (i,j=1,"',n). We get following equation from above relation.. ( -Oh. lhN-tAtB-BE,tl AtD-Btc>. O1-X BtC-AtD CtD-DtC1. -(:Si )( i2 ,`S) -(A. B. )( -O. h, )( i2 ,C tD. ). Theorem 5.1 (Modular Transformation) Let put period matrix for a base : At,Bt( i,j= 1, "' ,n ) on n =( H, ll'), and period matrixfor a base : 'A t,'Bt( i,j= 1,'",n ), on " st =( 'n, ' H'), then we getfollowing equationjbr two base which satishp the equation('). (a) "n=st ( iBA ICD) (b)*T.. C+DT A+BT. Proof.. * z,]=.IIA, coi=. h. nl.1(a･jlA+bv･IBt). " 7r'u=./IB, cc)i= ./s) nt.i(Cu'tA'du'tBl). nn l=1 l=1 nn. n. wi" 2a,,.L,,tu' 2hi L, to- 2ajinit+. w'= 2c,tJl,,tu' 2dJt. A, to. (" n, H ') =(n 71+ fi tB. l=1. " tc+ n ED). (IBA ICD). whichproves(a)by'T=H9A+n'13, 'n`=C+nCD *T=(n {A+ ll ,tB )"i( fi ℃+ H TeD). =(n({A+n'ifi'tB)l'i{I[(tc+n'iHT{D)}. =(za+TCB)-i(tC+TD) by Tand *T are symmetric matrix,. l=1. n n = 2c]i7rit+ 2biin,i. l=1. l=1･. (*n,ll')=(Hnt). l=1. n. 2loJt7r,t. l=1.

(22) 32. Seiji YAMAzAKI, Minoru ITo, Yoshimitu SAITo, Toshikazu IKEDA, Masahiko KuBo, Masaaki TAsHIRo, Teiichi HIGucHI. t(* T )= l( EA+ T tB )"i(tC+ T CD )l. *T=t(tC+TtD)t({A+TtB)-i 'T=(C+t(TtD))(A+t(TtB)l-i "T= (C+DT )( A+B T )-i which proves (b). 5.2 Concrete example n=2. We show a concrete example of translation S. Namely, we choose two base of homology for the Riemann's surface of y2=x6- 1. The following figure show it. Two base. ts;. rt-xB2. - Bl. ･t. --v. A. -d. -"d. rc R. N" A2 ,. Bl e-. n 6' s.. fi l. --dt -!. "'"""""f. A2. Al. .""v-. llSA ･･. --- ee. "". eee. B2. N- -e"･. ee. *A,. ". IS1. ,),cEid. '. ee. of homology'l Ai, A2, Bi, B2l and-( "Ai, 'A2, "Bi, "B2l has next relation clearly.. (*Al )-(s g･ s,,g × il) *A2 *Bl *B2. by result of 3.2 and theorem 5.1,. (*ll n')=(ll ll')( za tC. IB.).

(23) RIEMANN'S PERIoD MATRIx oF y2=x2n+2-1. 33. -(fi fi･)(gi･ ¥-a). oo OIL )(ft--lt .ft:Pil}:P.i,Zft.?))(ww) = 2K(2)( P2-P3 ( k-Ps Pi-P32 R?-IIIZ. *ll.2. (5i O)(Ri-A Ri-Ig. )(l ?)-2K(2)(ft:.A,,:i.ilZ,P,i, lt:,i,il,). K> P?-P,2 Re-R?. * fi '= 2K( 2) (. Pi-P2 Pi-P2-P3-P4 )(6 -,i)-2K2)(&':,P,2 .P,i:S P,2-P P,2-P-P?-R?. *T= 2(P+P2) (. P;P2. ). ).viii(i l)t. P+P2 2(P+P2) 5.1.2 g2=x2n+2-1. Likewise y2=x6- 1, we think two base of homology for y2=x2n+2 - 1. The figure show it.. sih)gB l,s. '.eet -" -.de -ee -p A? esN A3. ts. A2. Al. -"-ee. .. Bl. '. An-1. "-. et. An. An-i An. ,. .. i. ,. ee. B2. !i D - e.. '. .ee. "b. -t. A3. l. '. Bn. Bno. }IB;25". r-}isBs}i;. -b e. B3. bee"ee. .. B. Bn-. n. '.

(24) Seiji YAMAzAKI, Minoru ITo, Yoshimitu SAITo, Toshikazu IKEDA.Masahiko KuBo, Masaaki TAsHIRo, Teiichi HIGucHI. 34. ta1. e2. --s--e. .. *Bi. Bl. .. *B2. ;. *B. wsvrf:. 'B. *Ai ll<l.*Atl･ ..illiA) 's..1.......e.....1`ltfcS/ iilllllt:i.::･.'.IC:{l,a.,.,,,,".'.:.'....').e.':i:>;,. Sg -e----se-b---"ee--t"--"e--"s--e--e-"---e"t---"e"teee --)--ee-"eeeb-e-"-ee"e---"e-s--e-et--eee"-e-eee"ee--b----ee-eV. Two base of homology-[Ai,'", An, Bi,'''Bn] and["Ai,''', "An, 'Bi,"''Bn] y2=x2n+1-1. are related by modular transformation S.. 1.. *A, Ai il 'An *Bl= An Bl 'Bn il Bn o. o. .. .. -. 1. .. 1. o. 1. 1. .. o. *n=n (o 1. ''. ''. l )=2K(n)(. .. .. 1. pi-p2 ･･･ 27., R,,-, - Bt. :. :. )(6 ･･:l). pr- p; ･ 27.,P;,-, -P;,. 'Ht. fi ( 1. 1. .o1 ･'. --. )=2K(n)( P2-P3''' R2n-P2n+J P:: Pg "' P;n -:H P:n+i. )(1 ･= 9). for.

(25) RIEMANN'S PERIOD MATRIX OF y2.x2n+2-1. "fl= (" n. ,'. fi ') for a basc of homology -[ "A i,'", "An, 'B.･･･, 'Bh].. 2K(n)( Pi-R, L-R, P7- P; p;- pt. 'H=. 2K(n)( *ll'=. 35. -P7- P: P;- PZ --. Bn-i - Bn ---. P22n;l ' P22n. i ---. P2in..i - P2en.. 2ni.i(Li - P2tJi) '" .R?n2- P>n-i + an - Rn+i 2 ,".,(Pi - P,2,-,). l 2 :.i(P,"i - P2"i-i). ). --. Rn - Bn+i. "' Pin2- Pii-i + P￡t- Pi+i. P22n - PA+i. " PZn -2 - PZi -i + P5n - P2nn+i. P2n. - P2".+i. l. ). From above calculation, we get new period matrix 'n=(" H, ll ') for the Riemann's surface of y2=x2n+2-1. Let prove that this new period matrix satisfy Riemann's period relation 2.1. By ' II[ 'ij = "A, cvi=2Kt 2 tn.･ (P5i ' PSi+i) ' II 'ij ,= f･B, coi=2Ki (PSt-i - PSt). nnn. ,2.,[ f,,toif.,tojl- ,2.,[ 2(lli-i-Pg,)Kl2 i-, (R{-pS,.,)Kl]. -4Ki K} 'fiS -Illi, P`tr- ,2n., Pg,..,,Pi', l=O. which satisfy lemma 2.1. Next,. nnn. ,2.,[ 11,,,toi+./L,evi]- ,Iil],[ 2(p-S,.,-pl,)K) 2 ,2., (pt,',-pt,',.,)Ki). n .4Kf (.1-Pt.) '. 2 [1-p5n-k"i)t] R(1+Pt) k.i (1-pi)(1-Pt)(n+1) = 4K2. t 1(1+Pi)l2. im kEi[ flAk dii JlkBk tuj] = l(iliill.2' )1, (n+i)sin( .2 .n i )> o.

(26) 36 SeijiYAMAzAKI,MinoruITo,YoshimituSAITo,ToshikazuIKEDA,MasahikoKuBo,MasaakiTAsHIRo,TeiichiHIGucHI. which proves lemma 2.2. From the above mentioned, we got two period matrix n and * n for the Riemann surface of y2=x2n+2-1 andn ,' fl satisfy the Riemann period relation.. Still morest and'n are related by modular transformation S. If we choose the other modular transformations S'-1S'1= 1, we will many other Riemann's period matrix. As we. got period for a fundamental base of homology on Riemann surface of y2=x2n+2-1, we can get value of period for all 1'cycle. In this way we get next Theorem.. Theorem 5.2 Letput Uon integral valueforafixed integralpathforRiemann suJface Rn of y2=x2n+2-1 then the integralvaluefor any integralpath C on Rn are given hyfollowing.. .L, 7:xSlli: i dx=U+ ,2".i[ 2ak ii (P2t-i-P2,)KL}-2b,(p,,-p,,.,)K]. therefore,. a,,b,Ez,p-exp( nn+tl)K=fee 7=x:l=1 dX. References [1] Tashiro, Yamazaki, Higuchi:Determinant Period Matrix of Hyperelliptic Riemann's Surface Bull.Fac.Gen.Ed. Tokyo University Agri. Pech. No 31,(1994).. [2] Yamazaki, Ito, Ikeda, Kubo, Higuchi:Riemann's Period Matrix of y4=x4-1 Inprinted from Jounal of the Yokohama Nationa} University SEC.1 No 41 (1994).. [3] Ito, Higuchi:Riemann's Period Matrix of y2=x2n'i-1 Master Article of Yokohama National University (1995).. [ 4 ] Divid Mamford:Tata Lectures on Theta 1 Containing Introduction and motovation :theta function in one variable Basic results on theta functions inseveral variables. Birkhauser (1983).. [5] Raghavan Narasimhan:Compact Riemann Surfaces Birkhauser (1992). [6] KoutarouOikawa:RiemannSurface ili.;5Zt±IHIfi(1987). [7] Tanzou Takeuchi:Elliptic Function Theory (1956).. [8]SirouAndou:IntroductiontoEllipticIntegralandEllipticFunction HjkEt±lllll (1987)..

(27) RIEMANN，S PERIOD MATRIX OF Y2・X2n＋2−1 37. ［9］. A．Hurwitz， R．Courant：Elliptic Function Theoryシュプリンガー・フェアラーク東京（1ggi）．．. ［10］. Ahlfors：Comprex Analysis現代数学社（1982）．. ［11］. H．Cartan：Complex Function Theory．岩波書店（1972）．. ［12］. Yukio Kusunoki：Function Theory朝倉書店（19与6）．. ［13］. Jirou Tamura：Analytic Function 裳華房書店（1983）．. ［14］. Makoto Nanba：Geometry of入rgebraic Curve 現代数学社（1991），.

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