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

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

So an element of ${\displaystyle {\tilde {\mathbf {E} }}^{+}}$ is a sequence ${\displaystyle (x_{1},x_{2},\ldots )}$ of elements ${\displaystyle x_{i}\in {\mathcal {O}}_{\mathbf {C} _{p}}/(p)}$ such that ${\displaystyle x_{i+1}^{p}\equiv x_{i}{\pmod {p}}}$. There is a natural projection map ${\displaystyle f:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbf {C} _{p}}/(p)}$ given by ${\displaystyle f(x_{1},x_{2},\dotsc )=x_{1}}$. There is also a multiplicative (but not additive) map ${\displaystyle t:{\tilde {\mathbf {E} }}^{+}\to {\mathcal {O}}_{\mathbf {C} _{p}}}$ defined by ${\displaystyle t(x_{,}x_{2},\dotsc )=\lim _{i\to \infty }{\tilde {x}}_{i}^{p^{i}}}$, where the ${\displaystyle {\tilde {x}}_{i}}$ are arbitrary lifts of the ${\displaystyle x_{i}}$ to ${\displaystyle {\mathcal {O}}_{\mathbf {C} _{p}}}$. The composite of ${\displaystyle t}$ with the projection ${\displaystyle {\mathcal {O}}_{\mathbf {C} _{p}}\to {\mathcal {O}}_{\mathbf {C} _{p}}/(p)}$ is just ${\displaystyle f}$. The general theory of Witt vectors yields a unique ring homomorphism ${\displaystyle \theta :W({\tilde {\mathbf {E} }}^{+})\to {\mathcal {O}}_{\mathbf {C} _{p}}}$ such that ${\displaystyle \theta ([x])=t(x)}$ for all ${\displaystyle x\in {\tilde {\mathbf {E} }}^{+}}$, where ${\displaystyle [x]}$ denotes the Teichmüller representative of ${\displaystyle x}$. The ring ${\displaystyle \mathbf {B} _{dR}^{+}}$ is defined to be completion of ${\displaystyle {\tilde {\mathbf {B} }}^{+}=W({\tilde {\mathbf {E} }}^{+})[1/p]}$ with respect to the ideal ${\displaystyle \ker \left(\theta :{\tilde {\mathbf {B} }}^{+}\to \mathbf {C} _{p}\right)}$. The field ${\displaystyle \mathbf {B} _{dR}}$ is just the field of fractions of ${\displaystyle \mathbf {B} _{dR}^{+}}$.

• 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