Jacobi Determination of Endomorphism Type of Jacobian Varieties of Hyperelliptic Curves over Finite Fields Kazuto MATSUO, Jinhui CHAO, and Shigeo TSUJ
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1 Jacob Determnaton of Endomorphsm Type of Jacoban Varetes of Hyperellptc Curves over Fnte Felds Kazuto MATSUO, Jnhu CHAO, and Shgeo TSUJII CM CM CM lftng CM CM lftng Jacob Jacob ordnary Jacob Jacob Frobenus 1. Jacob Jacob Toyo Communcaton Equpment Co., Ltd., 1 1 Koyato 2, Samukawa-mach, Koza-gun, Kanagawa-ken, Japan Dept. of Electrcal, Electronc, and Communcaton Engneerng, Faculty of Scence and Engneerng, Chuo Unversty, Kasuga, Bunkyo-ku, Tokyo, Japan Dept. of Informaton and System Engneerrng, Faculty of Scence and Engneerng, Chuo Unversty, Kasuga, Bunkyo-ku, Tokyo, Japan dvsor Dvsor [1] [5] [6] [20] CM 1 CM (2) [18], [20], [21] 2 (3) [8], [14], [17], [18] (1) theta [6], [7], [9], [10], [12], [13], [18], [19] [6], [12], [18], [19] CM A Vol. J85 A No. 6 pp
2 2002/6 Vol. J85 A No. 6 CM lftng [22], [23] CM theta CM [11], [15], [16] Jacob Jacob Kohel [24] Jacob CM lftng Jacob 2 2. Jacob CM order p 2 F q g C C : Y 2 = F (X) (1) F (X) =X 2g+1 + a 2gX 2g + + a 0 F q [X] (2) F (X) J C C Jacob J C F q -dvsor J C(F q ) g C/F q Jacob #J C(F q ) Hasse-Wel Range ( q 1) 2g #J C(F q ) ( q +1) 2g (3) J C(F q ) J C[n] J C n-torson group J C[p a ] = (Z/p a Z) r, 0 r g (4) r = g J C ordnary J C ordnary Jacob C ordnary C ordnary J C smple J C End(J C) K = Q ZEnd(J C) (5) J C CM [K : Q] =2g End(J C) K order O End(J C) lftng CM [11], [15], [16] C [24] π q J C q Frobenus χ q(x) Z[X] K = Q(π q ) (6) χ q(x) =X 2g s 1X 2g 1 + s 2X 2g 2 s 1q g 1 X + q g, s Z (7) [25]K maxmal order O K Z-bass rank 2g Z-module 1. K 2g CM O K K maxmal order O K Z-bass B K B K =(ω 1 ω 2 ω 2g ) (8) ω = f (π q)/d,f (X) Z[X], deg f = 1, d Z, d 1 d,f 1 =1,d 1 =1. 678
3 Jacob Proof. [26, V Lemma1.1] B K χ q(x) [26] [28] K B πq =(1 π q π 2g 1 q ) (9) B πq Z[π q] Z-bass BZ[π q] =(f 1(π q) f 2(π q) f 2g(π q)) (10) Z[π q] Z-bass 3. [24] Kohel [24] ordnary F q ordnary E End(E) E F q c conductor π q E q Frobenus K E CM conductor c Kohel O K Z-bass a, m Z B = 1 π q+a m c 2 (Kohel). n m ker(π q + a) E [n] End(E) Z+ Z πq + a n E m j c c = m Q p j Kohel j E dvson polynomal 4. Jacob Dvson polynomal Kohel Baby step gant step algorthm 4. 1 ordnary Jacob Jacob J C/F q End(J C) J C CM K order K order End(J C) = O E O K (11) O E O E Z-bass π q J C q Frobenus π q End(J C) (12) O E Z[π q] O E O K (13) 1 f m = Y p e B 0 =(f 1(π q) f 2g 1(π q) f 2g (π q ) q g 1 ) (14) p E E h h p j p j +1 ker(π q + a) ker(π q + a) B 0 K order O 0 Z-bass O 0 Z[π q] O 0 O K (15) O 0 679
4 2002/6 Vol. J85 A No O 0 O E (16) Proof. πq 1 B πq 0 1 b1 πq 1 = B πq C,b. A Q (17) b 2g (7) 0 q g b 2g π qπq 1 = B πq b 2g 1 + s 1b 2g 1 C A = 1 (18) b 2g = 1/q g 0 1 qb 1 qπq 1 = B πq 1 q g 1 C A (19) qπ 1 q O E (20) [29, Theorem 7.4] 1 f (16) O 0 O O K (21) K order O End(J C) O O E Algorthm 1 F q ordnary C Jacob J C Algorthm 1 step 1 J C q Frobenus π q χ q(x) step 8 O 0 O O K K order Z-bass B O step 10 ω O ω End(J C) J C/F q 4. 2 χ q F q C Jacob J C q Frobenus π q χ q(x) Z[X] (22) C zeta Z(X, C) d X dx log Z(X, C) = N X 1 (23) =1 N C F q - Z(X, C) χ q(x) Z(X, C) = X 2g χ q( 1 X ) (1 X)(1 qx) N = q +1 2gX j=1 (24) π ϕ j q (25) π ϕ j q,j =1...2g πq 2g χ q C F q - =1...g Algorthm 2 680
5 Jacob Algorthm 2 step 2 domnant part step 2 a F q g a (qg 1)/2 Algorthm 2 O(g 3 q g (log q) 3 ) (26) Algorthm 2 χ q(x) Elkes [30] Elkes O g 2 q 8g/5 /4 (log q) 2 (27) Algorthm 2 Elkes Algorthm 2 Elkes Algorthm Algorthm 6 J C/F q q n Frobenus π q n χ q n χ q n Algorthm 2 F q n Algorthm 3 χ q n Algorthm 3 step 4 R,j R j step 5 I 2g 2g 2g Algorthm 3 g,n domnant part step 5 π q π q = q (28) χ q n(x) =X 2g s 1X 2g 1 + s 2X 2g 2 s 1q (g 1)n X + q gn (29) s Z, =1...g (28) s ψ 2g! q g 2 < 2 2g q gn 2 (30) Hessenberg [28] step 5 O((2g) 3 (log(2 2g q gn )) 3 )=O(g 6 n 3 (log q) 3 ) (31) (31) Algorthm O Algorthm 1 O 0 O O K O Z-bass Z[π q] O O K O Z-bass (8) B K (10) BZ[π q] O Z-bass 681
6 2002/6 Vol. J85 A No Z[π q] OBZ[π q] BZ[π q] = B OA O Z-bass B O A = (a j) Z 2g 2g a a d a j( < j) 0 a j <a. [27, 3 Lemma(2.9)()] A Z 2g 2g a j( < j) 0 a j <a det A 0 O O K B O = B KA 1 A 1 =(b j) Z 2g 2g BZ[π q] = B KA 2 A 2 =(c j) Z 2g 2g c = d A 1 Q 2g 2g A 1 = A 2A 1 A 1 A 2 = A 1A A 1A a b = d a d 4 O O O K B O = B K Ā (32) Ā Ā Z 2g 2g (33) AZ[π q] Z 2g 2g, A K, A O Q 2g 2g Q(π q ) B πq BZ[π q] = B πq AZ[π q] (34) B K = B πq A K (35) B O = B πq A O (36) 4 (32) AZ[π q] = A OA (37) A O = A K Ā (38) 4 A Ā = A 1 K AZ[π q]a 1 Q 2g 2g (39) Z O Z[π q] O O K Z[π q] O O K O Z-bass O O O 0 A 2g 2g a 2g,2g q g 1 a 2g,2g Algorthm 4 O 0 O O K O Z-bass Algorthm 4 step 5 A (8) d c = 2gY =1 d (40) d d c 1/(2g+1 ) a 2g,2g O(q 1 g d 2g) <2g, j a j O(d ) 2g +1 a j A #{A} = O(q 1 g c 2g ) A Ā ĀA = A 1 K AZ[π q] (41) A 1 K A Z[π q] d A a j c 1/(2g+1 ) Ā O(g(log c) 2 ) Algorthm 4 O(gq 1 g c 2g (log c) 2 ) (42) χ q(x) dscrmnant χ q(x) q dsc (χ q) (4q) g(2g 1) (43) 682
7 Jacob p c dsc (χ q) Algorthm 4 O(2 2g2 (2g 1) g 5 q 2g3 g 2 g+1 (log q) 2 ) (44) 4. 4! 2O! 2 End(J C ) Algorthm 1 step 10 ω O O K ω End(J C) 4. 3 O Z[π q] J C q Frobenus π q ω O π q ω = f(π q)/d, d Z,f(X) Z[X] (45) ω f,d (45) 5 5. ω = f(π q)/d O f(x) Z[X],d Z ω End(J C) J C[d] ker f(π q) (46). ω End(J C) D J C[d] dd =0 ωdd =0 f(π q)d =0 J C[d] ker f(π q) σ 1 : dj C J C/J C[d] (47) Can. σ 2 : J C/J C[d] J C/ ker f(π q) (48) σ 3 : J C/ ker f(π q) f(π q)j C (49) σ 1 dj C = J C J C/J C[d] σ 2 J C/J C[d] J C/ ker f(π q) σ 3 J C/ ker f(π q) f(π q)j C = J C σ 2 σ ω : J C J C (50) D σ 3(σ 2(σ 1(D))) (51) D J C dω(d) =ω(dd) =f(π q)d (52) ω = f(π q)/d 5 ω End(J C) dvsor D J C[d] f(π q)d = 0 (53) ω End(J C) 6. l d s d l part d = Y s,s = l e (54) s, J C[s ] ker f(π q) J C[d] ker f(π q) (55) 7. d Z G =(D 1 D 2 D 2g ) (56) J C[d] J C[d] =ZD 1 + ZD ZD 2g (57) dvsor D J C[d],f(π q)d =0 D G, f(π q)d =0 (58) d part s J C[s] G dvsor D G (53) ω End(J C) Algorthm 5 ω O ω End(J C) Algorthm 5 s = l e J C[s] G G Cantor [31] dvson polynomal J C q Frobenus χ q(x) 683
8 2002/6 Vol. J85 A No. 6 πq n χ q n(x) Algorthm 3 #J C(F q n )=χ q n(1) s #J C(F q n ) J C[s](F q n ) φ J C[s](F q n ) φ F q n l e 0 #J C(F q n ) e 0 s 0 = l e 0 D J C(F q n ) D = #JC(Fqn ) s 0 D J C[s 0] (60) D D J C[s 0](F q n ) dvsor D Baby step gant step algorthm [28], [32], [33] J C[s 0](F q n ) G 0 G 0 =( D1 D2 D2g ) (61) # D = lẽ ē ê = max{0, ē e} G =(lê1 D1 lê2 D2 lê2g D2g ) (62) J C[s](F q n ) s J C[s] G Algorthm 6 Algorthm 6 Algorthm 6 step 3 Algorthm 3 (31) n O(g 6 n 3 (log q) 3 ) gcd(q,s) =1 n =1...O(s 2g ) χ q n #J C(F q n ) Algorthm 6 O(g 6 s 8g (log q) 3 ) (63) gcd(q,s) 1 n =1...O(s g ) χ q n O(g 6 s 4g (log q) 3 ) (64) n G 0 (3) #J C(F q n )=O(q gn ) (65) Baby step gant step algorthm D J C[l e 0 ](F q n ) O(gn log q) dvsor D O(g) D O(g 4 (n log q) 3 ) (66) Baby step gant step algorthm O( l e 0 ) dvsor O( l e 0 (gnlog q) 2 ) (67) 684
9 Jacob e 0 = O(g log l n) Baby step gant step algorthm O(g 2 l g n 5/2 (log q) 2 ) (68) gcd(q,s) =1 1 n<o(s 2g ) G 0 n O( g l) step 6 Algorthm 6 G 0 D J C[l e 0 ](F q n ) O(g 4 s 6g (log q) 3 ) (69) Baby step gant step algorthm O(g 2 l g s 5g (log q) 2 ) (70) l s O(g 4 s 6g (log q) 3 ) (71) gcd(q,s) 1 1 n<o(s g ) G 0 πqd,=1...2g s gcd(q,s) =1 O(g 3 s 4g (log q) 2 log s ) (77) gcd(q,s) 1 O(g 3 s 2g (log q) 2 log s ) (78) 2g D f(π q)d s gcd(q,s) =1 O(g 3 s 4g (log q) 2 (log q + g log s )) (79) gcd(q,s) 1 O(g 3 s 2g (log q) 2 (log q + g log s )) (80) s = s (73) (74) Algorthm 5 domnant part Algorthm 6 Algorthm 5 d l part s gcd(q,s) =1 O(g 4 s 3g (log q) 3 ) (72) O(g 6 (log q) 3 X s 8g ) (81) Algorthm 6 gcd(q,s) =1 O(g 6 s 8g (log q) 3 ) (73) gcd(q,s) 1 O(g 6 s 4g (log q) 3 ) (74) Algorthm 5 Algorthm 5 step 3 D G f(π q)d π qd, =1...2g s gcd(q,s) =1 O(g 2 s 4g (log q)3 ) (75) gcd(q,s) 1 O(g 2 s 2g (log q)3 ) (76) gcd(q,s) 1 O(g 6 (log q) 3 s 4g ) (82) 5. Algorthm 1 Jacob Algorthm 5 domnant part Algorthm 6 Algorthm 6 Algorthm 1 torson Algorthm 1 Algorthm 6 (40) c d 1 = d 2 = = d 2g 1 =1, d 2g = c = O((4q) g(2g 1) 2 ) (83) case Algorthm 6 685
10 2002/6 Vol. J85 A No. 6 J C q Frobenus π q χ q(x) dscrmnant q g(g 1) dsc (χ q)=q g(g 1) d case d 2g = O p dsc (χq) q = O q g(g 1) d (84) (81) (82) Algorthm 6 q part (81) q = p n d 2g p part p g(g 1)n/2 part O( d ) part part s gcd(q,s) =1 s = O(2 g(2g 1) q g2 /2 ) (85) gcd(q,s) 1 3 s = O(q (g 1)(g 2)/2 ) (86) (73) (74) gcd(q,s) =1 domnant O(2 8g2 (2g 1) g 6 q 4g3 (log q) 3 ) (87) (44) (87) Algorthm 1 6. q 2 2 Magma [34] χ q Elkes [30] Algorthm 6 step 8 [28, Algorthm 5.4.1] Baby step gant step algorthm Algorthm 6 step 10 #G <4 Algorthm 5 G G Algorthm 5 step Pentum III 866 MHz memory 500 MByte p {7, 17, 31, 61, 127, 251, 509, 1031} F p Jacob memory (40) c Algorthm 6 n N 1 1 #{C } memory 1 p = c N Jacob c O(p 2 ) N O(p) c = O(q 2 ) (88) N = O(q) (89) (42) g =2 (88) Algorthm 4 686
11 Jacob 1 Table 1 Expermental results of computng endomorphsm rngs of hyperellptc curves over F p. p #{C } Memory (MByte) c N C Table 2 Lst of curves that endomorphsm rngs of ther Jacoban varetes could not be determned n the experment (Denote as C ). p C c N 127 Y 2 = X 5 +98X 4 +75X 3 +97X 2 +32X Y 2 = X X X 3 +58X 2 +99X Y 2 = X X X X X Y 2 = X X X 3 +10X X Y 2 = X X 4 +55X X X Y 2 = X X X X X O(q 7 (log q) 2 ) (90) (89) Algorthm 5 Algorthm 6 step 3 Algorthm 3 (31) n =1...N χ q n O(q 4 (log q) 3 ) (91) D (66) O((q log q) 3 ) (92) Baby step gant step algorthm (68) l = q O(q 9/2 (log q) 2 ) (93) (90) p = 1031 F C C : Y 2 = X X 4 +47X X X (94) C Jacob J C p Frobenus π p χ p(x) =X 4 +45X X X (95) J C CM K = Q(π p ) (96) maxmal order O K Z-bass B K =(ω 1 ω 2 ω 3 ω 4 ), (97) ω 1 =1, ω 2 = π q, ω 3 = π2 q +5π q +2, 7 ω 4 = π3 q πq π q O 0 Z-bass 687
12 2002/6 Vol. J85 A No. 6 B 0 =(ω 1 ω 2 7ω 3 14ω 4 ) (98) O 0 O O K K order O 8 O O O E J C[2] ker f 1(π p) (99) J C[7] ker f 2(π p) (100) J C[7] ker f 3(π p) (101) f 1(X) =X (102) f 2(X) =X 3 +5X 2 +2X (103) f 3(X) =X 2 +5X + 2 (104) J C[2] #J C(F 1031 n ) n =1, 2,... n =1 #J C(F 1031) = (105) J C[2] J C(F 1031) φ J C[2](F 1031) J C[2](F 1031) = Z/2Z (106) D f 1(π p)d = 0 (107) #J C(F 1031 n ) n =2 #J C(F )= (108) J C[2] J C(F ) φ J C[2](F ) J C[2](F ) = (Z/2Z) 2 (109) f 1(π p)d 0 (110) D J C (99) f 1(π q) 2 / End(J C) (111) J C[7] #J C(F 1031 n ) n =24 #J C(F )=7 8 N 24 (112) J C[7] J C (F ) φ N N bt J C[7](F ) J C[7](F ) = (Z/7Z) 2 (113) 2 D f 2(π p)d = 0 (114) f 3(π p)d = 0 (115) #J C(F 1031 n ) n = 168 #J C(F )=7 8 N 168 (116) J C[7] J C(F ) φ N N bt J C[7](F ) J C[7](F ) = (Z/7Z) 4 (117) f 2(π p)d 1 0 (118) f 3(π p)d 2 0 (119) D 1, D 2 J C (100) (101) f 2(π q) / End(J C) 7 (120) f 3(π q) / End(J C) 7 (121) End(J C) = O 0 (122) 3 36 MByte memory J C[7](F ) 688
13 Jacob 3 Table 3 Tmng of determnng the endomorphsm rng of the hyperellptc curve (94). χ p 0.1 O s.t O 0 O O K 0.05 #J C(F 1031 n ) 4.0 J C [2](F 1031) 0.01 J C [2](F ) 0.04 J C[7](F ) 7.0 J C[7](F ) (99) 0.00 (100) 2.2 (101) lftng CM ordnary Jacob (87) lftng CM q 1000 F q 2 memory Kohel 3. [24] [1] D.G. Cantor, Computng n the Jacoban of hyperellptc curve, Math. Comp., vol.48, no.177, pp , [2] S. Paulus and A. Sten, Comparng real and magnary arthmetcs for dvsor class groups of hyperellptc curves, ANTS-III, no.1423 n Lecture Notes n Computer Scence, pp , Sprnger-Verlag, [3] P. Gaudry and R. Harley, Countng ponts on hyperellptc curves over fnte felds, ANTS-IV, ed. W. Bosma, no.1838 n Lecture Notes n Computer Scence, pp , Sprnger-Verlag, [4] K. Nagao, Improvng group law algorthms for Jacobans of hyperellptc curves, ANTS-IV, ed. W. Bosma, no.1838 n Lecture Notes n Computer Scence, pp , Sprnger-Verlag, [5] K. Matsuo, J. Chao, and S. Tsuj, Fast genus two hyperellptc curve cryptosystems, IEICE Techncal Report, ISEC , [6] A.M. Spallek, Kurven vom Geshlcht 2 und hre Anwendung n Publc-Key-Kryptosystemem, PhD thess, GH Essen, [7] X. Wang, 2-dmensonal smple factors of J 0(N), Manuscrpta Math., vol.87, no.2, pp , [8] K. Matsuo, J. Chao, and S. Tsuj, Desgn of cryptosstems based on Abelan varetes over extenson felds, IEICE Techncal Report, ISEC97-30, [9] H.J. Weber, Hyperellptc smple factor of J 0(N) wth dmenson at least 3, Expermental Math., vol.6, no.4, [10] G. Frey and M. Müller, Arthmetc of modular curves and applcatons, n Algorthmc algebra and number theory, ed. B. Matzat, G. Greuel, and G. Hss, pp.11 48, Sprnger-Verlag, [11] K. Matsuo, J. Chao, and S. Tsuj, On lftng of CM hyperellptc curves, Proc. of SCIS 99, pp , [12] P.V. Wamelen, Example of genus two CM curves defned over the ratonals, Math. Comp., vol.68, no.225, pp , [13] P.V. Wamelen, Provng that a genus 2 curve has complex multplcaton, Math. Comp., vol.68, no.228, pp , [14] J. Chao, K. Matsuo, and S. Tsuj, Fast constructon of secure dscrete logarthm problems over Jacoban varetes, Informaton Securty for Global Informaton Infrastructures: IFIP TC 11 16th Annual Workng Conference on Informaton Securty, ed. S. Qng and J.Eloff, pp , Kluwer Academc Pub., [15] J. Chao, K. Matsuo, H. Kawashro, and S. Tsuj, Constructon of hyperellptc curves wth CM and ts applcaton to cryptosystems, Advances n Cryp- 689
![2002/6 Vol. J85 A No. 6 tology - ASIACRYPT2000, ed. T. Okamoto, no.1976 n Lecture Notes n Computer Scence, pp.259 273, Sprnger-Verlag, 2000. [16] Ordnary lftng CM Proc. of SCIS2000, no.c51, 2000.](/docs-images/94/118385202/images/14-0.jpg "[17] CM Wel number Proc. of SCIS2000, no.c50, 2000. [18] A. Weng, Constructng hyperellptc curves of genus 2 sutable for cryptography, preprnt, 2000. [19] 2 Proc. of SCIS2001, pp.749 754, 2001.")
![[20] A vol.j84-a, no.8, pp.1045 1053, Aug. 2001. [21] J.F. Mestre, Constructon de curbes de genere 2 à partr de leurs modules, n Effectve methods n algebrac geometry, ed. C. T. T. Mora, no.](/docs-images/94/118385202/images/14-1.jpg "94 n Progress n Mathematcs, pp.313 334, Brkhäuser, 1991. [22] J. Chao, O. Nakamura, K. Sobataka, and S. Tsuj, Constructon of secure ellptc cryptosystems, Advances n Cryptology - ASIACRYPT 98, ed. K. Ohta and D.")
![Pe, no.1514 n Lecture Notes n Computer Scence, pp.95 109, Sprnger-Verlag, 1998. [23] CM A vol.j82-a, no.8, pp.1261 1268, Aug. 1999. [24] D.](/docs-images/94/118385202/images/14-2.jpg "Kohel, Endomorphsm rngs of ellptc curves over fnte felds, PhD thess, UCB, 1996. [25] H. Stchtenoth, Algebrac functon felds and codes, Unverstext, Sprnger-Verlag, 1993. [26] M.")
14 2002/6 Vol. J85 A No. 6 tology - ASIACRYPT2000, ed. T. Okamoto, no.1976 n Lecture Notes n Computer Scence, pp , Sprnger-Verlag, [16] Ordnary lftng CM Proc. of SCIS2000, no.c51, [17] CM Wel number Proc. of SCIS2000, no.c50, [18] A. Weng, Constructng hyperellptc curves of genus 2 sutable for cryptography, preprnt, [19] 2 Proc. of SCIS2001, pp , [20] A vol.j84-a, no.8, pp , Aug [21] J.F. Mestre, Constructon de curbes de genere 2 à partr de leurs modules, n Effectve methods n algebrac geometry, ed. C. T. T. Mora, no.94 n Progress n Mathematcs, pp , Brkhäuser, [22] J. Chao, O. Nakamura, K. Sobataka, and S. Tsuj, Constructon of secure ellptc cryptosystems, Advances n Cryptology - ASIACRYPT 98, ed. K. Ohta and D. Pe, no.1514 n Lecture Notes n Computer Scence, pp , Sprnger-Verlag, [23] CM A vol.j82-a, no.8, pp , Aug [24] D. Kohel, Endomorphsm rngs of ellptc curves over fnte felds, PhD thess, UCB, [25] H. Stchtenoth, Algebrac functon felds and codes, Unverstext, Sprnger-Verlag, [26] M. Pohst, Computatonal Algebrac Number Theory, no.21 n DMV Semner, Brkhäuser, [27] M. Pohst and H. Zassenhaus, Algorthmc Algebrac Number Theory, no.30n Encyclopeda of mathematcs and ts applcatons, Cambrdge U.P., [28] H. Cohen, A Course n Computatonal Algebrac Number Theory, no.138 n Graduate Text n Mathematcs, Sprnger-Verlag, [29] W.C. Waterhouse, Abelan varetes over fnte felds, Ann. scent. Ec. Norm. Sup., 4 o t. 2, pp , [30] N.D. Elkes, Ellptc and modular curves over fnte felds and related computatonal ssues, n Computatonal perspectves on number theory, ed. D.A. Buell and J.T. Tetlbaum, pp.21 76, AMS, [31] D.G. Cantor, On the analogue of the dvson polynomals for hyperellptc curves, Journal für de rene und angewandte Mathematk, vol.447, pp , [32] H. Cohen, Advanced Topcs n Computatonal Number Theory, no.193 n Graduate Text n Mathematcs, Sprnger-Verlag, [33] E. Teske, A space effcent algorthm for group structure computaton, Math. Comp., vol.67, pp , [34] D IEEE 690
62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P
![62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P](/thumbs/91/106734463.jpg "62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P")
15, pp.61-80 Abel-Jacob I 1 Introducton Remann Abel-Jacob X g Remann X ω 1,..., ω g Λ = {( γ ω 1,..., γ ω g) C g γ H 1 (X, Z)} Λ C g lattce Jac(X) = C g /Λ Le Abel-Jacob (Theorem 2.2, 4.2) Jac(X) Pcard
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, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n
( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally