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I have some questions concerning period rings for Galois representations.

First, consider the case of $p$-adic representations of the absolute Galois group $G_K$, where $K$ denote a $p$-adic field. Among all these representations, we can distinguish some of them, namely those which are Hodge-Tate, de Rham, semistable or crystalline. This is due to Fontaine who constructed some period rings : $B_{HT}$, $B_{dR}$, $B_{st}$ and $B_{crys}$.

Constructing the ring $B_{HT}$ is not very difficult and it is quite natural.

Does someone have any idea where $B_{dR}$ comes from ?

For $B_{crys}$, I guess it was constructed to detect the good reduction of (proper, smooth ?) varieties. I don't know anything of crystalline cohomology but does someone have a simple explanation of the need to use the power divided enveloppe of the Witt vectors of the perfectisation (?) of $\mathcal{O}_{\mathbb{C}_p}$ ?

As for the ring $B_{st}$, once you have $B_{crys}$, I think the idea of Fontaine was to add a period from Tate's elliptic curve, which have bad semistable reduction. Does someone knows if Fontaine was aware that adding just this period will be sufficient or was it a good surprise ?

Finally, why there is no period rings for global $p$-adic Galois representations ?

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Laurent Berger has written a very readable introduction to p-adic Hodge theory in the "Geometric aspects of Dwork theory" volume; you can find it on his web page at umpa.ens-lyon.fr/~lberger/article05/article05.pdf. That has some useful motivation for the constructions. –  David Loeffler Feb 7 '11 at 23:30
    
I found Fontaine's paper "Le corps des periodes p-adiques" in Asterisque pretty useful. Here he gives a different construction of the period rings/fields. In fact he constructs them as universal objects in the right categories. –  Arijit Aug 15 '12 at 21:35

2 Answers 2

Beilinson has recently discovered a new proof of the de Rham comparison isomorphism. You can find a write-up here: http://arxiv.org/abs/1102.1294. Here, he shows that $B_{dR}$ naturally shows up when you consider the p-adic completion (in a suitable sense) of the derived de Rham cohomology of $\mathcal{O}_{\bar{K}}$ over $\mathcal{O}_K$.

Also, $A_{cris}$ naturally shows up as (more or less) the global sections of the structure sheaf over the crystalline site for $\mathcal{O}_{\bar{K}}$ over $W(k)$ ($k$ is the residue field of $K$). There is a very nice explanation of this in R. S. Lodh's thesis: http://www.math.utah.edu/~remi/research/thesispt1formatted.pdf.

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Beilinson's results (two papers, one mentioned by Keerthi and the other here) have been generalised by Bhargav Bhatt; his paper also introduces a global period ring $A_{ddR}$ for global Galois representations!! The ring $A_{ddR}$ is a filtered $\hat{Z}$-algebra equipped with a Gal$(\bar{Q}/{Q})$-action.

A (one of the many) beautiful result in this paper is the following theorem:

Let $X$ be a semistable variety over $Q$. Then the log de Rham cohomology of a semistable model for $X$ is isomorphic to the $\hat{Z}$-etale cohomology of $X_{\bar{Q}}$, once both sides are base changed to a localization of $A_{ddR}$ (whle preserving all natural structures on either side).

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Good find! How did you come across it? –  Keerthi Madapusi Pera Feb 4 '12 at 2:58
    
Thanks! I was just browsing the Geometric Langlands page for their recent activities: math.uchicago.edu/seminars/geometric_langlands.html –  SGP Feb 4 '12 at 22:29
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I should say that the preprint above is an old version of a privately circulated draft (thanks to Rob Harron for pointing this post out). There are surely some yet-to-be-fixed inaccuracies/gaps, and the material in the last section, which is referred to in the post, will be fleshed out more in the future. –  Bhargav Feb 4 '12 at 23:00

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