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 ?
Best Answer
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.