The automorphism groups of Kummer surfaces associated with the product of two elliptic curves
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- by Jonghae Keum and Shigeyuki Kondō PDF
- Trans. Amer. Math. Soc. 353 (2001), 1469-1487 Request permission
Abstract:
We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.References
- Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI 10.1016/0021-8693(87)90245-6
- 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
- J. H. Conway, The automorphism group of the $26$-dimensional even unimodular Lorentzian lattice, J. Algebra 80 (1983), no. 1, 159–163. MR 690711, DOI 10.1016/0021-8693(83)90025-X
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, 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, DOI 10.1007/978-1-4757-2016-7
- Jong Hae Keum, Automorphisms of Jacobian Kummer surfaces, Compositio Math. 107 (1997), no. 3, 269–288. MR 1458752, DOI 10.1023/A:1000148907120
- J. H. Keum, Every algebraic Kummer surface has infinitely many automorphisms, unpublished manuscript (1996).
- Shigeyuki Kond\B{o}, Enriques surfaces with finite automorphism groups, Japan. J. Math. (N.S.) 12 (1986), no. 2, 191–282. MR 914299, DOI 10.4099/math1924.12.191
- Shigeyuki Kond\B{o}, The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geom. 7 (1998), no. 3, 589–609. MR 1618132
- S. Kondō, The maximum order of finite groups of automorphisms of $K3$ surfaces, Amer. J. Mathematics 121 (1999), 1245–1252.
- 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, DOI 10.1007/BF01388829
- V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
- 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
- I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI 10.2969/jmsj/02410020
- T. Shioda and H. Inose, On singular $K3$ surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR 0441982
- J. A. Todd, A representation of the Mathieu group $M_{24}$ as a collineation group, Ann. Mat. Pura Appl. (4) 71 (1966), 199–238. MR 202854, DOI 10.1007/BF02413742
- 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
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
- MR Author ID: 291447
- Email: jhkeum@kkucc.konkuk.ac.kr, jhkeum@kias.re.kr
- Shigeyuki Kondō
- Affiliation: Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
- Email: kondo@math.nagoya-u.ac.jp
- 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.
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1806732