Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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), 1469-1487
MSC (2000): Primary 14J28, 14J50, 11H56
DOI: https://doi.org/10.1090/S0002-9947-00-02631-3
Published electronically: September 13, 2000
MathSciNet review: 1806732
Full-text PDF

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.


References [Enhancements On Off] (What's this?)

  • [1] R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133-153. MR 89b:20018
  • [2] J. H. Conway, Three lectures on exceptional groups, in Finite simple groups, Academic Press, 1971, pp. 215-247. MR 49:2918
  • [3] J. H. Conway, The automorphism group of the $26$ dimensional even Lorentzian lattice, J. Algebra 80 (1983), 159-163. 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), 269-288. 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  • [6] J. H. Keum, Every algebraic Kummer surface has infinitely many automorphisms, unpublished manuscript (1996).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), 191-282. MR 89c:14058
  • [8] S. Kondo, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geometry 7 (1998), 589-609.MR 99i:14043
  • [9] S. Kondo, The maximum order of finite groups of automorphisms of $K3$ surfaces, Amer. J. Mathematics 121 (1999), 1245-1252.
  • [10] S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on cohomology groups, Invent. math. 77 (1984), 383-397. MR 86i:14012
  • [11] V.V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type $K3$, Moscow Math. Soc. 38 (1980), 71-137. MR 81e:32033
  • [12] V.V. Nikulin, Integral symmetric bilinear forms and their applications, Math. USSR Izv. 14 (1980), 103-167. MR 80j:10031
  • [13] I. Piatetski-Shapiro, I.R. Shafarevich, A Torelli theorem for algebraic surfaces of type $K3$, Math. USSR Izv. 5 (1971), 547-587. MR 44:1666
  • [14] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20-59. MR 55:2927
  • [15] T. Shioda, H. Inose, On singular $K3$ surfaces, Complex analysis and algebraic geometry (Collection Dedicated to K. Kodaira), Iwanami Shoten, Tokyo, and Cambridge Univ. Press, Cambridge (1977), 119-136. MR 56:371
  • [16] J.A. Todd, A representation of the Mathieu group $M_{24}$ as a collineation group, Ann. Mat. Pure Appl. 71 (1966), 199-238. 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. 323-348. 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 143-701, Korea and Korea Institute for Advanced Study, Seoul 130-012, Korea
Address at time of publication: Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea
Email: jhkeum@kkucc.konkuk.ac.kr, jhkeum@kias.re.kr

Shigeyuki Kondo
Affiliation: Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Email: kondo@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-00-02631-3
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(1999-2-102-002-3). The second author was supported in part by the Monbusho Grant-in Aid for Scientific Research (B) 10440005 and Houga 11874004.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society