P-adic Teichmüller theory: Difference between revisions
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In mathematics, '''p-adic Teichmüller theory''' describes the |
In [[mathematics]], '''''p''-adic Teichmüller theory''' describes the "uniformization" of [[p-adic|''p''-adic]] curves and their [[Moduli_spaces|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|first=Shinichi|last=Mochizuki|year1=1996|year2=1999}}. |
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The first problem is to reformulate the Fuchsian [[Uniformization theorem|uniformization]] of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the 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 ''p''-adic Teichmüller theory, the ''p''-adic analogue the Fuchsian uniformization of Teichmüller theory, is the study of integral Frobenius invariant indigenous bundles. |
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==See also== |
==See also== |
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*[[Anabelian geometry]] |
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*[[nilcurve]] |
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==References== |
==References== |
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| ⚫ | *{{Citation | last1=Mochizuki | first1=Shinichi | title=A theory of ordinary p-adic curves | doi=10.2977/prims/1195145686 |mr=1437328 | year=1996 | journal=Kyoto University. Research Institute for Mathematical Sciences. Publications | issn=0034-5318 | volume=32 | issue=6 | pages=957–1152| doi-access=free | hdl=2433/59800 | hdl-access=free }} |
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| ⚫ | *{{Citation | last1=Mochizuki | first1=Shinichi | title=Foundations of p-adic Teichmüller theory | url=https://books.google.com/books?id=JnMWs5BLTiEC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=AMS/IP Studies in Advanced Mathematics | isbn=978-0-8218-1190-0 |mr=1700772 | year=1999 | volume=11}} |
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*{{Citation | last1=Mochizuki | first1=Shinichi | editor1-last=Berthelot | editor1-first=Pierre | editor1-link=Pierre Berthelot (mathematician) | editor2-last=Fontaine | editor2-first=Jean-Marc | editor3-last=Illusie | editor3-first=Luc | editor3-link=Luc Illusie | editor4-last=Kato | editor4-first=Kazuya | editor4-link=Kazuya Kato | editor5-last=Rapoport | editor5-first=Michael | title=Cohomologies p-adiques et applications arithmétiques, I. |mr=1922823 | year=2002 | journal=Astérisque | issn=0303-1179 | issue=278 <!-- | chapter=An introduction to p-adic Teichmüller theory | chapter-url=http://www.kurims.kyoto-u.ac.jp/~motizuki/An%20Introduction%20to%20p-adic%20Teichmuller%20Theory.pdf --> | pages=1–49}} |
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{{DEFAULTSORT:P-adic Teichmuller theory}} |
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| ⚫ | *{{Citation | last1=Mochizuki | first1=Shinichi | title=A theory of ordinary p-adic curves |
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| ⚫ | *{{Citation | last1=Mochizuki | first1=Shinichi | title=Foundations of p-adic Teichmüller theory | url= |
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[[Category:Algebraic geometry]] |
[[Category:Algebraic geometry]] |
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[[Category:Number theory]] |
[[Category:Number theory]] |
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[[Category:p-adic numbers]] |
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Latest revision as of 01:29, 27 June 2024
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 Shinichi Mochizuki (1996, 1999).
The first problem is to reformulate the Fuchsian uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the 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 p-adic Teichmüller theory, the p-adic analogue the Fuchsian uniformization of Teichmüller theory, is the study of integral Frobenius invariant indigenous bundles.
See also
[edit]References
[edit]- 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, hdl:2433/59800, ISSN 0034-5318, MR 1437328
- 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, MR 1700772
- 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, MR 1922823
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