Journal of Algebraic Geometry

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		Online ISSN 1534-7486; Print ISSN 1056-3911

Non-Archimedean orbifolds covered by Mumford curves

Author(s): Fumiharu Kato
Journal: J. Algebraic Geom. 14 (2005), 1-34.
Posted: July 20, 2004
MathSciNet review: 2092125
Retrieve article in: PDF

Abstract | References | Additional information

Abstract: In this article we develop a theory of the so-called Mumford orbifolds, viz. rigid-analytic orbifolds covered by Mumford curves. General recipe for treating such orbifolds is given. The main result states a necessary and sufficient condition for abstract graphs of groups to be realized as discrete groups for Mumford orbifolds, which is useful for constructing several interesting non-Archimedean discrete groups, such as -adic triangle groups.

References:

1.: André, Y.: -adic orbifolds and -adic triangle groups, RIMS Kyoto proceedings (Surikaisekikenkyusho Kokyuroku) No. 1073, proceedings of the conference ``rigid geometry and group action'' Kyoto, December 1998, 136-159. MR 1713695 (2000g:14032)
2.: André, Y., Baldassarri, F.: De Rham cohomology of differential modules on algebraic varieties, Progress in Mathematics 189, Birkhäuser Verlag, Basel, 2001. MR 1807281 (2002h:14031)
3.: Cornelissen, G., Kato, F., Kontogeorgis, A.: Discontinuous groups in positive characteristic and automorphisms of Mumford curves, Math. Ann. 320 (2001), 55-85. MR 1835062 (2003d:14032)
4.: Cornelissen, G., Kato, F.: Mumford curves with maximal automorphism group, Proc. Amer. Math. Soc. 132 (2004), 1937-1941. MR 2053963
5.: Cornelissen, G., Kato, F., Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic, Duke Math. J. 116 (2003), 431-470. MR 1958094 (2004c:14044)
6.: Gerritzen, L., van der Put, M.: Schottky groups and Mumford curves, Lecture Notes in Math., 817, Springer, Berlin, 1980. MR 0590243 (82j:10053)
7.: Herrlich, F.: Über Automorphismen -adischer Schottkykurven, Dissertation, Bochum, 1978.
8.: Herrlich, F.: Endlich erzeugbare -adische diskontinuierliche Gruppen, Arch. Math. 35 (1980), 505-515. MR 0604249 (82f:14022)
9.: Herrlich, F.: Die Ordnung der Automorphismengruppe einer -adischen Schottkykurve, Math. Ann. 246 (1980), 125-130. MR 0564682 (81f:14015)
10.: Herrlich, F.: -adisch diskontinuierlich einbettbare Graphen von Gruppen, Arch. Math. 39 (1982), 204-216. MR 0682448 (84d:14013)
11.: Ishida, M.-N., Kato, F.: The strong rigidity theorem for non-Archimedean uniformization, Tôhoku Math. J. 50 (1998), 537-555. MR 1653430 (2000a:14028)
12.: Kato, F.: -adic Schwarzian triangle groups of Mumford type, preprint, 2000.
13.: Kato, F.: Mumford curves in a specialized pencil of sextics, Manuscripta Math. 104 (2001), 451-458. MR 1836106 (2002c:14045)
14.: Katz, N.: Exponential sums and differential equations, Annals of Mathematics Studies 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536 (93a:14009)
15.: Maskit, B.: Kleinian groups, Grundl. der math. Wiss. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988. MR 0959135 (90a:30132)
16.: Mumford, A.: An analytic construction of degenerating curves over complete local rings, Compositio Math. 24, Fasc. 2 (1972), 129-174. MR 0352105 (50:4592)
17.: Mustafin, G. A.: Nonarchimedean uniformization, Math. USSR Sbornik 34 (1978), 187-214.
18.: Reni, M.: A graph-theoretical approach to Kleinian groups, Proc. London Math. Soc. (3) 67 (1993), 200-224. MR 1218126 (94b:57043)
19.: Reversat, M.: Sur les revêtements de Schottky des courbes modulaires de Drinfeld, Arch. Math. 66 (1996), 378-387. MR 1383902 (97e:11073)
20.: Serre, J-P.: Trees, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
21.: van der Put, M.: The structure of $\Omega$ and its quotients $\Gamma\backslash\Omega$ , in Proceedings of the Workshop on ``Drinfeld Modules, Modular Schemes and Applications'' (Gekeler, E.-U., van der Put, M., Reversat, M., Van Geel, J. ed.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1997, 103-112. MR 1630594 (99b:11002)
22.: Weber, H.: Lehrbuch der Algebra, Chelsea Publishing Company, New York, 1899.
23.: Yoshida, M.: Fuchsian differential equations, Aspects of Mathematics Vol. E11, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1987.

Additional Information:

Fumiharu Kato
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: kato@math.kyoto-u.ac.jp
DOI: 10.1090/S1056-3911-04-00384-4
PII: S 1056-3911(04)00384-4
Received by editor(s): April 12, 2002
Received by editor(s) in revised form: November 4, 2003
Posted: July 20, 2004

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