P-adic Teichmüller theory: Difference between revisions
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{{lowercase|''p''-adic Teichmüller theory}} |
{{lowercase|''p''-adic Teichmüller theory}} |
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In mathematics, '''p-adic Teichmüller theory''' describes the "uniformization" of [[p-adic|''p''-adic]] curves and their moduli, generalizing the usual [[Teichmüller theory]] that describes the [[Uniformization theorem|uniformization]] of [[Riemann surfaces]] and their moduli. It was introduced and developed by {{harvs|txt|authorlink=Shinichi Mochizuki|last=Mochizuki|year1=1996|year2=1999}}. |
In mathematics, '''p-adic Teichmüller theory''' describes the "uniformization" of [[p-adic|''p''-adic]] curves and their [[Modulus pace|moduli]], generalizing the usual [[Teichmüller theory]] that describes the [[Uniformization theorem|uniformization]] of [[Riemann surfaces]] and their moduli. It was introduced and developed by {{harvs|txt|authorlink=Shinichi Mochizuki|last=Mochizuki|year1=1996|year2=1999}}. |
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The first problem is to reformulate the Fuchsian uniformization of a complex Riemann surface in a way that makes sense for ''p''-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical [[indigenous bundle]] over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose monodromy representation is quasi-Fuchsian. For ''p''-adic curves the analogue of complex conjugation is the [[Frobenius endomorphism]], and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So the ''p''-adic analogue of the Fuchsian uniformization of Teichmüller theory is the study of indigenous bundles that are invariant under the Frobenius element and also integral. |
The first problem is to reformulate the Fuchsian [[Uniformization theorem|uniformization]] of a complex Riemann surface in a way that makes sense for ''p''-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical [[indigenous bundle]] over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose [[monodromy]] representation is quasi-Fuchsian. For ''p''-adic curves the analogue of complex conjugation is the [[Frobenius endomorphism]], and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So the ''p''-adic analogue of the Fuchsian uniformization of Teichmüller theory is the study of indigenous bundles that are invariant under the [[Frobenius element]] and also integral. |
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==See also== |
==See also== |
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Revision as of 13:34, 7 October 2013
In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Mochizuki (1996, 1999).
The first problem is to reformulate the Fuchsian uniformization of a complex Riemann surface in a way that makes sense for p-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical indigenous bundle over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose monodromy representation is quasi-Fuchsian. For p-adic curves the analogue of complex conjugation is the Frobenius endomorphism, and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So the p-adic analogue of the Fuchsian uniformization of Teichmüller theory is the study of indigenous bundles that are invariant under the Frobenius element and also integral.
See also
References
- Mochizuki, Shinichi (1996), "A theory of ordinary p-adic curves", Kyoto University. Research Institute for Mathematical Sciences. Publications, 32 (6): 957–1152, doi:10.2977/prims/1195145686, ISSN 0034-5318, MR1437328
- Mochizuki, Shinichi (1999), Foundations of p-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1190-0, MR1700772
- Mochizuki, Shinichi (2002), Berthelot, Pierre; Fontaine, Jean-Marc; Illusie, Luc; Kato, Kazuya; Rapoport, Michael (eds.), "Cohomologies p-adiques et applications arithmétiques, I.", Astérisque (278): 1–49, ISSN 0303-1179, MR1922823
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