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 [math]\displaystyle{ \mathbf{B}_{dR} }[/math] is defined as follows. Let [math]\displaystyle{ \mathbf{C}_p }[/math] denote the completion of [math]\displaystyle{ \overline{\mathbf{Q}_p} }[/math]. Let

[math]\displaystyle{ \tilde{\mathbf{E}}^+ = \varprojlim_{x\mapsto x^p} \mathcal{O}_{\mathbf{C}_p}/(p) }[/math]

So an element of [math]\displaystyle{ \tilde{\mathbf{E}}^+ }[/math] is a sequence [math]\displaystyle{ (x_1,x_2,\ldots) }[/math] of elements [math]\displaystyle{ x_i\in \mathcal{O}_{\mathbf{C}_p}/(p) }[/math] such that [math]\displaystyle{ x_{i+1}^p \equiv x_i \pmod p }[/math]. There is a natural projection map [math]\displaystyle{ f:\tilde{\mathbf{E}}^+ \to \mathcal{O}_{\mathbf{C}_p}/(p) }[/math] given by [math]\displaystyle{ f(x_1,x_2,\dotsc) = x_1 }[/math]. There is also a multiplicative (but not additive) map [math]\displaystyle{ t:\tilde{\mathbf{E}}^+\to \mathcal{O}_{\mathbf{C}_p} }[/math] defined by [math]\displaystyle{ t(x_,x_2,\dotsc) = \lim_{i\to \infty} \tilde x_i^{p^i} }[/math], where the [math]\displaystyle{ \tilde x_i }[/math] are arbitrary lifts of the [math]\displaystyle{ x_i }[/math] to [math]\displaystyle{ \mathcal{O}_{\mathbf{C}_p} }[/math]. The composite of [math]\displaystyle{ t }[/math] with the projection [math]\displaystyle{ \mathcal{O}_{\mathbf{C}_p}\to \mathcal{O}_{\mathbf{C}_p}/(p) }[/math] is just [math]\displaystyle{ f }[/math]. The general theory of Witt vectors yields a unique ring homomorphism [math]\displaystyle{ \theta:W(\tilde{\mathbf{E}}^+) \to \mathcal{O}_{\mathbf{C}_p} }[/math] such that [math]\displaystyle{ \theta([x]) = t(x) }[/math] for all [math]\displaystyle{ x\in \tilde{\mathbf{E}}^+ }[/math], where [math]\displaystyle{ [x] }[/math] denotes the Teichmüller representative of [math]\displaystyle{ x }[/math]. The ring [math]\displaystyle{ \mathbf{B}_{dR}^+ }[/math] is defined to be completion of [math]\displaystyle{ \tilde{\mathbf{B}}^+ = W(\tilde{\mathbf{E}}^+)[1/p] }[/math] with respect to the ideal [math]\displaystyle{ \ker\left( \theta : \tilde{\mathbf{B}}^+ \to \mathbf{C}_p \right) }[/math]. The field [math]\displaystyle{ \mathbf{B}_{dR} }[/math] is just the field of fractions of [math]\displaystyle{ \mathbf{B}_{dR}^+ }[/math].

References

• Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory, I, Berlin: Walter de Gruyter GmbH & Co. KG, ISBN 978-3-11-017478-6, Bibcode: 2002math.....10184B 
• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory, http://math.stanford.edu/~conrad/papers/notes.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, 223, Paris: Société Mathématique de France