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Fontaine's period rings

From Wikipedia, the free encyclopedia

In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc Fontaine[1] that are used to classify p-adic Galois representations.

The ring BdR

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The ring is defined as follows. Let denote the completion of . Let

So an element of is a sequence of elements such that . There is a natural projection map given by . There is also a multiplicative (but not additive) map defined by , where the are arbitrary lifts of the to . The composite of with the projection is just . The general theory of Witt vectors yields a unique ring homomorphism such that for all , where denotes the Teichmüller representative of . The ring is defined to be completion of with respect to the ideal . The field is just the field of fractions of .

Notes

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  1. ^ Fontaine (1982)

References

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  • Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory, vol. I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, Bibcode:2002math.....10184B, ISBN 978-3-11-017478-6, MR 2023292
  • Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05
  • Fontaine, Jean-Marc (1982), "Sur Certains Types de Representations p-Adiques du Groupe de Galois d'un Corps Local; Construction d'un Anneau de Barsotti-Tate", Ann. Math., 115 (3): 529–577, doi:10.2307/2007012
  • Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque, vol. 223, Paris: Société Mathématique de France, MR 1293969