Motivation of the construction of $p$-adic period rings

Question

Let $B$ be either $B_{\text{dR}}$ or $B_{\text{crys}}$. For a $\mathbb{Q}_p$-representation $V$ of the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of a $p$-adic field $K$ (a finite extension of $\mathbb{Q}_p$), there are subspaces of $H^1(K,V)$ defined by $\ker(H^1(K,V)\to H^1(K,B_\bullet\otimes V))$ for $\bullet\in\{\text{dR},\text{crys}\}$. Each element of the subspaces of $H^1(K,V)$ correponds of an extension of $\mathbb{Q}_p$ by $V$ in the category of $\mathbb{Q}_p$-representations of $\mathrm{Gal}(\overline{K}/K)$ having some nice properties -- being de Rham/crystalline. The constructions of the rings $B$ seem quite complicated. My question is

How did we end up with the such complicated constructions of $B$'s so that elements of subspaces of $H^1$ having such representation-theoretic properties? Or, why do elements in $\ker(H^1(K,V)\to H^1(K,B\otimes V))$ have such representation theoretic properties?

Answer 1

These rings of periods were constructed by Fontaine in the 70's and 80's, based on earlier work of Tate (on $p$-divisible groups). The constructions are indeed quite complicated, and it's all the more remarkable that they cut out the right categories. These rings did not, however, come out of nowhere. For instance, $B_{cris}$ is $H^0_{cris}(O_{\overline{Q}_p}/p.O_{\overline{Q}_p})$. There are similar interpretations for other rings of periods.

You should look at Fontaine's early papers on the subject, especially the "Périodes p-adiques" Astérisque volume, now available for free online. There are more recent surveys and course notes, for instance Caruso's https://hal.archives-ouvertes.fr/hal-02268787/document

Answer 2

How did we end up with the such complicated constructions of $B$?

To add to Laurent's answer remark that "these rings did not, however, come out of nowhere", I believe that in the early 80s, Fontaine noticed that two seemingly very different constructions naturally lead to very similar (classes of) rings:

1. The Honda systems he had introduced in his classification of $p$-divisible groups.

2. Certain surjective evaluation maps and functorial properties in the theory of the field of norms he had established with Wintenberger.

Once you have both, it is quite natural to look at the completion of some Witt vector rings with respect to the kernel of the evaluation maps in question, and that's $B_{\operatorname{dR}}^{+}$. You can have a look at the references given in the following answer.

https://mathoverflow.net/a/342838/2284

The fact that an extension of Galois representation by a de Rham (or crystalline, or...) representation corresponds to a class in $$\ker\left(H^1(G_K,V)\longrightarrow H^1(G_K,V\otimes B_{\operatorname{dR},\operatorname{crys},\dots})\right)$$ follows formally from the notion of $B$-admissibility: an extension $\xi$ of $\mathbb Q_p$ by $V$ is an extension by a $B$-admissible representation if and only if it becomes split after applying $D_{B}(\cdot)$ if and only if the class of $\xi$ is in the kernel of
$$\ker\left(H^1(G_K,V)\longrightarrow H^1(G_K,V\otimes B)\right).$$