Fontaine's period rings

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

The ring B_dR

The ring $\mathbf {B} _{dR}$ is defined as follows. Let $\mathbb {C} _{p}$ denote the completion of ${\overline {\mathbb {Q} _{p}}}$. Let

${\tilde {\mathbf {E} }}^{+}=\varprojlim _{x\mapsto x^{p}}{\mathcal {O}}_{\mathbb {C} _{p}}/(p).$

An element of ${\tilde {\mathbf {E} }}^{+}$ is a sequence $(x_{1},x_{2},\ldots )$ of elements $x_{i}\in {\mathcal {O}}_{\mathbb {C} _{p}}/(p)$ such that $x_{i+1}^{p}\equiv x_{i}\!\!\!{\pmod {p}}$. There is a natural projection map $f:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbb {C} _{p}}/(p)$ given by $f(x_{1},x_{2},\dotsc )=x_{1}$. There is also a multiplicative (but not additive) map $t:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbb {C} _{p}}$ defined by

$t(x_{,}x_{2},\dotsc )=\lim _{i\to \infty }{\tilde {x}}_{i}^{p^{i}}$,

where the ${\tilde {x}}_{i}$ are arbitrary lifts of the $x_{i}$ to ${\mathcal {O}}_{\mathbb {C} _{p}}$. The composite of $t$ with the projection ${\mathcal {O}}_{\mathbb {C} _{p}}\to {\mathcal {O}}_{\mathbb {C} _{p}}/(p)$ is just $f$.

The general theory of Witt vectors yields a unique ring homomorphism $\theta :W({\tilde {\mathbf {E} }}^{+})\to {\mathcal {O}}_{\mathbb {C} _{p}}$ such that $\theta ([x])=t(x)$ for all $x\in {\tilde {\mathbf {E} }}^{+}$, where $[x]$ denotes the Teichmüller representative of $x$. The ring $\mathbf {B} _{dR}^{+}$ is defined to be completion of ${\tilde {\mathbf {B} }}^{+}=W({\tilde {\mathbf {E} }}^{+})[1/p]$ with respect to the ideal $\ker(\theta :{\tilde {\mathbf {B} }}^{+}\to \mathbb {C} _{p})$. Finally, the field $\mathbf {B} _{dR}$ is just the field of fractions of $\mathbf {B} _{dR}^{+}$.

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