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
The ring
is defined as follows. Let
denote the completion of
. Let
![{\displaystyle {\tilde {\mathbf {E} }}^{+}=\varprojlim _{x\mapsto x^{p}}{\mathcal {O}}_{\mathbf {C} _{p}}/(p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de8b398cee10a7debe39ec41f1e996baa1022b0)
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
References
- 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