Non-Archimedean orbifolds covered by Mumford curves
Author:
Fumiharu Kato
Journal:
J. Algebraic Geom. 14 (2005), 1-34
DOI:
https://doi.org/10.1090/S1056-3911-04-00384-4
Published electronically:
July 20, 2004
MathSciNet review:
2092125
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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 $p$-adic triangle groups.
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A-B01 André, Y., Baldassarri, F.: De Rham cohomology of differential modules on algebraic varieties, Progress in Mathematics 189, Birkhäuser Verlag, Basel, 2001.
CKK99 Cornelissen, G., Kato, F., Kontogeorgis, A.: Discontinuous groups in positive characteristic and automorphisms of Mumford curves, Math. Ann. 320 (2001), 55–85.
C-K00 Cornelissen, G., Kato, F.: Mumford curves with maximal automorphism group, Proc. Amer. Math. Soc. 132 (2004), 1937–1941.
C-K01 Cornelissen, G., Kato, F., Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic, Duke Math. J. 116 (2003), 431–470.
GvP80 Gerritzen, L., van der Put, M.: Schottky groups and Mumford curves, Lecture Notes in Math., 817, Springer, Berlin, 1980.
Her78 Herrlich, F.: Ăśber Automorphismen $p$-adischer Schottkykurven, Dissertation, Bochum, 1978.
Her80a Herrlich, F.: Endlich erzeugbare $p$-adische diskontinuierliche Gruppen, Arch. Math. 35 (1980), 505–515.
Her80b Herrlich, F.: Die Ordnung der Automorphismengruppe einer $p$-adischen Schottkykurve, Math. Ann. 246 (1980), 125–130.
Her82 Herrlich, F.: $p$-adisch diskontinuierlich einbettbare Graphen von Gruppen, Arch. Math. 39 (1982), 204–216.
I-K98 Ishida, M.-N., Kato, F.: The strong rigidity theorem for non-Archimedean uniformization, Tôhoku Math. J. 50 (1998), 537–555.
Kat00 Kato, F.: $p$-adic Schwarzian triangle groups of Mumford type, preprint, 2000.
Kat01 Kato, F.: Mumford curves in a specialized pencil of sextics, Manuscripta Math. 104 (2001), 451–458.
Katz Katz, N.: Exponential sums and differential equations, Annals of Mathematics Studies 124, Princeton University Press, Princeton, NJ, 1990.
Mas88 Maskit, B.: Kleinian groups, Grundl. der math. Wiss. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.
Mum72 Mumford, A.: An analytic construction of degenerating curves over complete local rings, Compositio Math. 24, Fasc. 2 (1972), 129–174.
Mus78 Mustafin, G. A.: Nonarchimedean uniformization, Math. USSR Sbornik 34 (1978), 187–214.
Ren93 Reni, M.: A graph-theoretical approach to Kleinian groups, Proc. London Math. Soc. (3) 67 (1993), 200–224.
Rev96 Reversat, M.: Sur les revêtements de Schottky des courbes modulaires de Drinfeld, Arch. Math. 66 (1996), 378–387.
Ser80 Serre, J-P.: Trees, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
vdP97 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.
Web99 Weber, H.: Lehrbuch der Algebra, Chelsea Publishing Company, New York, 1899 .
Yos87 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
Received by editor(s):
April 12, 2002
Received by editor(s) in revised form:
November 4, 2003
Published electronically:
July 20, 2004
