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

Published electronically:
September 13, 2000

MathSciNet review:
1806732

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Abstract | References | Similar Articles | Additional Information

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**, 10.1016/0021-8693(87)90245-6**[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****[3]**J. H. Conway,*The automorphism group of the 26-dimensional even unimodular Lorentzian lattice*, J. Algebra**80**(1983), no. 1, 159–163. MR**690711**, 10.1016/0021-8693(83)90025-X**[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, 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****[5]**Jong Hae Keum,*Automorphisms of Jacobian Kummer surfaces*, Compositio Math.**107**(1997), no. 3, 269–288. MR**1458752**, 10.1023/A:1000148907120**[6]**J. H. Keum,*Every algebraic Kummer surface has infinitely many automorphisms*, unpublished manuscript (1996).**[7]**Shigeyuki Kondō,*Enriques surfaces with finite automorphism groups*, Japan.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****[8]**Shigeyuki Kondō,*The automorphism group of a generic Jacobian Kummer surface*, J. Algebraic Geom.**7**(1998), no. 3, 589–609. MR**1618132****[9]**S. Kondo,*The maximum order of finite groups of automorphisms of**surfaces*, Amer. J. Mathematics**121**(1999), 1245-1252.**[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**, 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****[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****[13]**I. I. Pjateckiĭ-Šapiro 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****[14]**Tetsuji Shioda,*On elliptic modular surfaces*, J. Math. Soc. Japan**24**(1972), 20–59. MR**0429918****[15]**T. Shioda and H. Inose,*On singular 𝐾3 surfaces*, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR**0441982****[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****[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**

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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