The automorphism groups of Kummer surfaces associated with the product of two elliptic curves
Authors:
Jonghae Keum and Shigeyuki Kondo
Journal:
Trans. Amer. Math. Soc. 353 (2001), 14691487
MSC (2000):
Primary 14J28, 14J50, 11H56
Published electronically:
September 13, 2000
MathSciNet review:
1806732
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.
 [1]
Richard
Borcherds, Automorphism groups of Lorentzian lattices, J.
Algebra 111 (1987), no. 1, 133–153. MR 913200
(89b:20018), http://dx.doi.org/10.1016/00218693(87)902456
 [2]
J.
H. Conway, Three lectures on exceptional groups, Finite simple
groups (Proc. Instructional Conf., Oxford, 1969) Academic Press, London,
1971, pp. 215–247. MR 0338152
(49 #2918)
 [3]
J.
H. Conway, The automorphism group of the 26dimensional even
unimodular Lorentzian lattice, J. Algebra 80 (1983),
no. 1, 159–163. MR 690711
(85k:11030), http://dx.doi.org/10.1016/00218693(83)90025X
 [4]
J.
H. Conway and N.
J. A. Sloane, Sphere packings, lattices and groups,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 290, SpringerVerlag, New York, 1988.
With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R.
A. Parker, L. Queen and B. B. Venkov. MR 920369
(89a:11067)
 [5]
Jong
Hae Keum, Automorphisms of Jacobian Kummer surfaces,
Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752
(98e:14039), http://dx.doi.org/10.1023/A:1000148907120
 [6]
J. H. Keum, Every algebraic Kummer surface has infinitely many automorphisms, unpublished manuscript (1996).
 [7]
Shigeyuki
Kondō, Enriques surfaces with finite automorphism
groups, Japan. J. Math. (N.S.) 12 (1986), no. 2,
191–282. MR
914299 (89c:14058)
 [8]
Shigeyuki
Kondō, The automorphism group of a generic Jacobian Kummer
surface, J. Algebraic Geom. 7 (1998), no. 3,
589–609. MR 1618132
(99i:14043)
 [9]
S. Kondo, The maximum order of finite groups of automorphisms of surfaces, Amer. J. Mathematics 121 (1999), 12451252.
 [10]
Shigeru
Mukai and Yukihiko
Namikawa, Automorphisms of Enriques surfaces which act trivially on
the cohomology groups, Invent. Math. 77 (1984),
no. 3, 383–397. MR 759266
(86i:14012), http://dx.doi.org/10.1007/BF01388829
 [11]
V.
V. Nikulin, Finite groups of automorphisms of Kählerian
𝐾3 surfaces, Trudy Moskov. Mat. Obshch. 38
(1979), 75–137 (Russian). MR 544937
(81e:32033)
 [12]
V.
V. Nikulin, Integer symmetric bilinear forms and some of their
geometric applications, Izv. Akad. Nauk SSSR Ser. Mat.
43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
(80j:10031)
 [13]
I.
I. PjateckiĭŠapiro and I.
R. Šafarevič, Torelli’s theorem for algebraic
surfaces of type 𝐾3, Izv. Akad. Nauk SSSR Ser. Mat.
35 (1971), 530–572 (Russian). MR 0284440
(44 #1666)
 [14]
Tetsuji
Shioda, On elliptic modular surfaces, J. Math. Soc. Japan
24 (1972), 20–59. MR 0429918
(55 #2927)
 [15]
T.
Shioda and H.
Inose, On singular 𝐾3 surfaces, Complex analysis and
algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136.
MR
0441982 (56 #371)
 [16]
J.
A. Todd, A representation of the Mathieu group
𝑀₂₄ as a collineation group, Ann. Mat. Pura
Appl. (4) 71 (1966), 199–238. MR 0202854
(34 #2713)
 [17]
È.
B. Vinberg, Some arithmetical discrete groups in
Lobačevskiĭ\ spaces, Discrete subgroups of Lie groups
and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ.
Press, Bombay, 1975, pp. 323–348. MR 0422505
(54 #10492)
 [1]
 R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133153. MR 89b:20018
 [2]
 J. H. Conway, Three lectures on exceptional groups, in Finite simple groups, Academic Press, 1971, pp. 215247. MR 49:2918
 [3]
 J. H. Conway, The automorphism group of the dimensional even Lorentzian lattice, J. Algebra 80 (1983), 159163. MR 85k:11030
 [4]
 J. H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss. Bd 290, 2nd ed. 80 (1988). MR 89a:11067
 [5]
 J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), 269288. MR 98e:14039
 [6]
 J. H. Keum, Every algebraic Kummer surface has infinitely many automorphisms, unpublished manuscript (1996).
 [7]
 S. Kondo, Enriques surfaces with finite automorphism groups, Japanese J. Math. 12 (1986), 191282. MR 89c:14058
 [8]
 S. Kondo, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geometry 7 (1998), 589609.MR 99i:14043
 [9]
 S. Kondo, The maximum order of finite groups of automorphisms of surfaces, Amer. J. Mathematics 121 (1999), 12451252.
 [10]
 S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on cohomology groups, Invent. math. 77 (1984), 383397. MR 86i:14012
 [11]
 V.V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type , Moscow Math. Soc. 38 (1980), 71137. MR 81e:32033
 [12]
 V.V. Nikulin, Integral symmetric bilinear forms and their applications, Math. USSR Izv. 14 (1980), 103167. MR 80j:10031
 [13]
 I. PiatetskiShapiro, I.R. Shafarevich, A Torelli theorem for algebraic surfaces of type , Math. USSR Izv. 5 (1971), 547587. MR 44:1666
 [14]
 T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 2059. MR 55:2927
 [15]
 T. Shioda, H. Inose, On singular surfaces, Complex analysis and algebraic geometry (Collection Dedicated to K. Kodaira), Iwanami Shoten, Tokyo, and Cambridge Univ. Press, Cambridge (1977), 119136. MR 56:371
 [16]
 J.A. Todd, A representation of the Mathieu group as a collineation group, Ann. Mat. Pure Appl. 71 (1966), 199238. MR 34:2713
 [17]
 E.B. Vinberg, Some arithmetic discrete groups in Lobachevskii spaces, in Proc. Int. Coll. on Discrete Subgroups of Lie Groups and appl. to Moduli (Bombay 1973), Oxford Univ. Press, 1975., pp. 323348. MR 54:10492
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
14J28,
14J50,
11H56
Retrieve articles in all journals
with MSC (2000):
14J28,
14J50,
11H56
Additional Information
Jonghae Keum
Affiliation:
Department of Mathematics, Konkuk University, Seoul 143701, Korea and Korea Institute for Advanced Study, Seoul 130012, Korea
Address at time of publication:
Korea Institute for Advanced Study, 20743 Cheongryangridong, Dongdaemungu, Seoul 130012, Korea
Email:
jhkeum@kkucc.konkuk.ac.kr, jhkeum@kias.re.kr
Shigeyuki Kondo
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 4648602, Japan
Email:
kondo@math.nagoyau.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994700026313
PII:
S 00029947(00)026313
Keywords:
Automorphisms of Kummer surfaces,
Picard lattice,
Leech lattice
Received by editor(s):
May 30, 1999
Received by editor(s) in revised form:
July 12, 1999
Published electronically:
September 13, 2000
Additional Notes:
The first author was supported by KOSEF(199921020023). The second author was supported in part by the Monbusho Grantin Aid for Scientific Research (B) 10440005 and Houga 11874004.
Article copyright:
© Copyright 2000
American Mathematical Society
