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(2) S. YAMAzAKi, M. ITo, T. IKEDA, M. KuBo and T. HiGucHi. 14. where I (7, 6) denotes the intersection number of 2 cycles 6 and 7,. We show these standard basis in the figure fig. 1. Cutting out the surface along these paths, we get a simply connected domain, which is the fig. 2.. el. 63. 6r. e2. rl+ 03 61+. rl-. /. e4. 012. ri. d3+ elo e'9 0ii rl. r3. es. e3. 010. e2 es. t. eg. 62. r2. 61. e6. 63-. O" ei-e4e6 e7 s. es. r3+. 62-. e12'. ;. ell. r2+. 62+ e7. r3- es. Fig. 2. Fig. 1. t P4. , P3.r ;. tt. -I. l. IV. lt. .･9-. "･' P3,. III. II. ''. "'"`' i ''! {N- : i)P2 t-- ･-･. 1. :. tl. :: ;i. P4. ll. I. I. ,. III b,. 'II :. N. ' Pi].;× tt x.J l ' ' 'tt s k vtti 'i x Pl N N. 'N, Pl Fig. 3. lt. r. Fig. 4. s 1 ,. N s s h. N. N N. IV.
(3) Riemann's Period matrix of y 4= x4- 1. 15. 2.3 Intergral paths of R3 The surface R3 is not hyperelliptic, and considered as a covering space of degree 4 over. a Riemann sphere. The fig. 3 is how the 6 cycles in fig. 2 are seen. From this figure, 6 integral paths 6i, 7i (i = 1, 2, 3) are drawn in fig. 4. Branching points are Pi = 1, P2. = i, P, = -1, P, = -i.. 61. 61. I. Pl. P3. P2. Pl. P4. I. rl. ..-pt-..---- )h.-b--"--..-". - rl P2. P3. -----･--. < ---------. P4 II. .v--ta- <---- 6i. rl. ---.---". 62. 62 -------･-----･----. II. Pl -----P2. rl. Sl rl. P4. P3. III. 62. r2. Pl. III. 82. II. l>. r2. 63. A-.----- )--.----------..- 63. -P2. '-'----v-- r-<. III'. ---=- r2 P3. P4. ---.-"L--<r-'-----' 63 ･-----< ----------. IV'. r2. (3 r2. IV 63. Fig. 5. 1 1. ------:>---- r3 ･------<r-- r3. Pl.P2. P3. Fig. 6. P4. IV.
(4) 16 S. YAMAzAKi, M. ITo, T. IKEDA, M. KuBo and T. HiGucHi 2.4 Abelian differencials on yP = xg-1 The set of all Abelian differentials on a Riemann surface of genus g forms a vector space, and the dimension of the vector space is equal to the genus g. The fact comes. from Riemann-Roch theorem. Theorem 1 in Particulany the dimension of the space on lhe aigebmic czarve yP = jeq1 is eqzaal to (P - 1) (q - 1)/2.. proof. The points where X2 drc may not be holomorphic are y = O or oo. At the points,. y. we seek an equality of a, b.. (i) y = O: From yP == xq- 1, pyP-' op == qxq-i du. Then du has a zero of order P - 1 at The points with y = O, that is, (x, y) == (ai, O), aiq - 1 = O (i = 1,..., q). Hence X.b. du is holomorphic at x = ai if. g. P-1}ta from an inequality (the order of zero points of xb dZt);}r(the order of zero points of y"). (ii)oo: We may assume the equation is yP == xq. Then, xbdu = ptmbP-P" dt and y" = t-aq. X.b dzv is holomorphic if. y - b2b ;}) aq +P+1 From the inequalities above a -- 1, 2,..., (p - 1) b = o, 1, 2,...,[aq - (pp + 1)] Hence, let N be the number of Abelian differentials of the form X.b du, then. N =- :2.l([ aq - SP + 1)]+ i) -- (p - i)iq - 1). Substituting P=q = 4, we have N == 3.. y.
(5) Riemann's Period matrix of y`=x`-1 17 Theorem 2 in fac4 holomomphic Abelian dip??rencials are following th7'ee fo7'ms:. y3, g3,y2. du xdu du '. proof. It is sufficient to check that the forms are holomophic at the branching points. x == 1, -1, i, -i and x = oo. At x = 1, let t= `fi. Then t` =1-x and 4t3 dO. = du Now, gy` = t`(t` + 2)(t` + 1 + i)(t` + - i), so. du 4dt. y3 - [(t`+ 2)(t`+1+ i)(t`+1- i)]i'. xdu- 4(t`+1)dt '. y3 - [(t4+2)(t4+1+i)(t4+1-i)]g'. dlx 4tdt. y2-[(t`+2)(t`+1+i)(t`+1-i)]}' . These are all holomorphic at t == O. So is it at x = -1, i, -i. At x = oo, let x = 1/t,. then. du -tdt xdlx -dt du -dt. g3-(1.t`)i'g3-(1-t4)i'g3-(lmt`)S . '. and those are holomorphic at t == O, i.e., at x == oo. These three Abelian differentials. are independent over C, and then form a basis of the vector space of all Abelian differentials on the Riemann surface defined by y` = x4 -1.. 3 Definitions Riemann's period matrix st for R3 is defined as the following matrix:. st-(-i')-(il ii 'kTiLi lil ieei)･. where 7vij = .,lgj cvi, 7rti,･ =: .IIj cvi(i, i = 1, 2, 3),. du xdu du. to1== 3)ca2== 3,to3== 2 y yy. When we integrate an Abelian differential on the paths, such as 6, 7 in Fig. 4, we must pay attention to the sign of the differentiaJs..
(6) 18. S. YAMAzAKi, M. ITo, T. IKEDA, M. KuBo and T. HiGucHi. Let toi, to2, to3 and ca4 be represented as caij(i = 1, 2, 3, 7' - 1, 2, 3, 4) on the 4 Riemann. surface in Fig. 4, then. Wll=(Vl Ct)21=ca2. to31 == W3. (V12=ZCVI CV22=Z(V2. to32 == -to3. W13=-to1 W23==-to2. CD33 = CV3. te14 == -ZWI ca24 == -Zto2. W34 = - ca3. 4 Period matrix of y`= x`-1 We calculate cai along the paths 61･s in Fig. I: 7qi = ZI, (vi = .Lif3 cvii + .Ll' (vi2 =: .Lif3 wi + .Ll;i(i(v,). = .4ir to, - .LIg cv, - i.Lir ca, + i.11r to, = (1 - i) (p, - p,) .('co w, zi2 === .LI, toi = .4If3 wi2 + .Li;i wi3 =: .11f3(iwi) + .Ll;i(- (v,). == .11r w, - .4r to, + i.4Ir to, - iy;; w, = (1 + i) (p, - p,) ,4co to, zi, - .11, to, - .LIf` .,, + .Lfi .,, -= ygf4(- .,) . .4ei(- ,.,). = - .11r toi + ylir to, + iL,co to, - i.Lir ca, = (-1 + i) (p, - p,) Ico to, 7121 =: .LI1 tu2 = .4;3 to21 + .LI;1 to22 = .,4If3 ca2 + .LI;i(ito2). = .LIr ca2 - .4; to2 - i.4r to2 + i.4; ca2 = (1 - i) (p12 - p32) 1['oo w2 n22 == .ZI, cv2 = .11f3 w22 + .Z;;i ca23 = .ZIIf3(ito2) + y4I;i(- a),). = .,4ir a), - .Li; w, + i.Lir w, - i.11; to, = (1 + i) (p,2 - p,2) .('oo to, 7r23 = .Zg, w2 = .LIf` to23 + .LI:i to,, = .LIf`(- w,) + .,4:'(m iw,). = .4Ir w2 + .4r cv2 + i.,(Ir to2 - i.Llr to2 :(-1 - i) (p12 - p32) v(1'oo ca2 nki == .Z;i ca3 == .Lf3 to3i + .Li;i tu32 = .LIf3 te3 + .111i(- to3). = yllr to3 - .igr, to, + yllr w, - .III to, == 2(p, - p,) ycco ca, nts2 = .Z;, cv3 = .LIf3 w32 + .11ei ca33 = .Lf3(- ca3) + .LIfi(to3). == -.Lr to, + .11; to, + .11g to, - y r to, = -2(p, - p,) .('co to,.
(7) Riemann's Period matrix of y`=x`- 1 nb3 = Z;3 w3 = .11f4 w33 + .Zl;i (v34 = .Lif4 cv3 + .Li:i(- to3). = L,co to3 - L,oo to3 + L,co ca3 - L,co to3 = 2(Pi - P4) Xco cv3. Let A, B and C be defined as follows: A = .11co to' == Ioo 4tillilir=i='B = .('co tu2 = u4co 4kdu 'C = .('oo W3= u('co tfliiii=. ,, .. (e&g,)(,1':,(,3')f({,;A-,,;Zl, ,Si±'s･ll,ve,i:)2, ,( ii',sl)/(,",i5;ll,). Substituting Pi = 1, P2 == i, P3 = -1 and Pi = 1, we have. ,, ...,(sw(('J, i) (it･,i' iii) IIIi - -8ABC (1 - i) In the following, we calculate the integrals in another representation of to'is. Zgi cai = .LI;3 wii + .Li;2 toi2 = .Lf3 wi + .LIge2(icai) == .4; a)i - i.4I; cvi - .11r cvi + i.4I; toi = (i - i) (p2 - p3) 11co (Di y(;, toi == y(I;3 wi2 + .J(I;2 toi3 = ./Ii;3(itoi) + .JC;2(- wi). = i.4: cvi + .11I; toi - i.Lir wi - .4ff toi = (i + i) (p2 - p3) ,4co wi. .Z;, wi = .Lf` wi3 + .11:i toi4 = .11f`(- wi) + .Li:i(- iw,). = -.4r to, + i.4r to, + .4Ir to, - iy;r w, = (-1+ i) (p, - p,) .,L'co to, Z;i to2 = .Li;3 to2i + .Lie2 to22 = .tl;3 w2 + .LI;2(ica2). = L,co to2 - iL,co to2 + iL,co to2 - iL,co to, = (1 - i) (p,2 - p,2) Ico to, .Zl]2 cv2 == .Li;3 cv22 + .11;2 cv23 = .4I;3(icv2) + u4I;2(- (v2). = i.1(l; to2 + .Ll; to2 - i.LIr to2 - .Lir to2 = a + i) (p22 - p32) y('ee w2 .[;3 to2 = .11f` w23 + .11:i to24 = .11f`(- toi) + .LI:i(- ito2). = - .LIr w2 + i.LIff ca2 + .Lr to2 - i.LIr to2 = (-1 + i) (p12 - p42) .('co cv2 ZI1 ca3 = .LI;3 to31 + .Li;2 to32 = yli;3 tu3 + .Llf2(- ca3). 19.
(8) 20 S. YAMAzAKi, M. ITo, T. IKEDA, M. KuBo and T. HiGucHi = .4I; to3 + .4i; cv3 - .Liff cv3 m ,lli: w3 = 2(p2 - p3) ,('co to3 .Z;, cv3 = .Li;3 w32 + .Li;2 ca33 = .L;3(- to3) + .4Ig2(ct)3). = -.L; to, - .4r cv, + .11; cv, + ,4Ir w, = -2(p, - p,) .ll'co cv, .ZI3 to3 = .11f4 to33 + .11:1 to34 = ,4f4 ca3 + .LIfl(' ca3). = Lico ca3 + L,co tu3 - L,co to3 - L,co to3 = 2(Pi - P4) fco to3. Let A, B and C be defined as follows: A - rfco wi -= fco 47 ge -r･B = Ioo w2 = Ico 47fffalIi･C= Ico ca3 =" foo 7 S:i ,, .. (e&g)(,1i:,(i')s(,-Pi,;l), 6i±+s･1:,;,ag･;,i!), ,( i;,sl)g(,ei,i;ll)). -2(e&g,)(-fill.i).,ai..I,, ,i,.l,,,). 1II1 =- 32ABC (1 + i) iiLi= i6ABci(i+i)(-('iiil/i.)i) il'i.i :8,±i))). Next, we calculate toi along the paths 7'js. n'ii = .IIi cai = .Li;4 cai2 + .LI:3 cvn == .41S4(itoi) + .Li:3 wi. - -.Li; ., + .Lr ., + i.Lig ., - i.LIr ., - (-i+ i) (p, - p,) ,4oo to, rr,i2 == .ll], w, = .11e` ca,, + .Lf3.,, == .L;4(-.,) + .LI:3(- i.,). = -.11; cvi + .4ir (v, + iyllgy cv, - i.4ir cv, = (-i + i) (p, - p,) .('co a), zt,, == I, to, == L,P2 to,, + L,Pi to,, = L,P2 ca, + L,Pi(- ito,). = .LIr to, - .11r w, + i.Lir to, - i.Lff to, == (1 + i) (p, - p,) .('co tu, z'2i == ,II]i w2 : .111` ca22 + .LI:3 cv2i = .LI;4(ito2) + .Lf3(w2). = - .LI; cv2 + .Lir cv2 + i.11; to2 - i.11r w2 = (-1 + i) (p32 - p42) .('oo to2 zt2, = .L], to, == .,4IS4 w,3 + .z;:3 cv,, = .LI;4(- w,) + .LI:3(- ito,).
(9) Riemann's Period matrix of y`=x`-1 21 = -L,oo ca2 + .LIff to2 + iL,co to2 h i.LIr to2 = (-1 + i) (p32 - p42) fco to2 z,23 =: ./I3 to2 = .4f2 to21 + .LIfl ca24 = .IIe2 tu2 + .Li;1(- ito2). == +L,co to2 ke Yli; to2 + iL,oo to2 - iYli; to2 = (1 + i) (P,2 - p,2) foo to,. n'3i = .[i to3 = .4;4 ca32 + .4:3 ca3i = L;4(- ca3) + y;:3(w3). = -L3co to3 + L,co to3 + L,co to3 - I,co tu3 = 2(Pi - P3) fco tu3 n'32 = y42 to3 == .4;4 cv33 + L:3 to34 = .4;4(to3) + ,4e3(-w3). = L,co to3 - L,co to3 + L,co ca3 - .LIr to3 = 2(p3 - p4) Ico to3 z'33 == YI3 to3 = .11if2 to3i + .Li;i w34 = .11f2 to3 + .Li;i(- w3). = L,co to3 - .LI; to3 + L,co ca3 - L,oo to3 = 2 (pi - p,) Ico to,. Let A, B and C be defined as follows:. A=. Ico toi == fco 47 f!=f･B=fco ca2=Ioo 4kdu ･C== fco ca3=Ico 7s{4=i. ii == (e&g)((L'il+-,(,3')s(p-P3;,:,,;1)) (L-ii+,sl)g(,eis;ll, 6ii(,)f(,-Pi,;il,). Substituting Pi = 1, P2 == i, P3 == -1 and Pi == 1, we have. ii･-2(6･&g)(it,i ,ii ii 5ii) 1IItl - 16ABC(1 - i). 1. L.
(10) 22 S. YAMAzAKi, M. ITo, T. IKEDA, M. KuBo and T. HiGucHi References [1] Y. Tashiro, S. Yamazaki and T. Higuchi, Determinant of Pen'od matn'ces of lmpe7elliptic. Riemann suhaces, Bull. Fac. Gen. Ed. Tokyo Univ. Agri. Tech. No. 31 (1994) (to appear).
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