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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, Fontain …

Consider the Tate motive $\mathbb Q(1)$. Its de Rham realization is simply $\mathbb Q$ (with the filtration $F^{-1}\mathbb Q=\mathbb Q$ and $F^{0}=0$) and its Betti realization is $2\pi i\mathbb Q$. T …

Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\math …

Fontaine's program is the classification of $p$-adic representations of $\operatorname{Gal}(\bar{K}/K)$ where $K$ is a discrete valuation field of residual characteristic $p$. If by motivation, you m …

Jean-Marc Fontaine Groupes p-divisibles sur les corps locaux. Astérisque 47-48, Soc. Math. France, Paris (1977), i+262 pp (especially chapter V) This is probably the canonical answer to your question …

The three articles referenced presented in logical order of exposition are respectively Faltings, Gerd Hodge-Tate structures and modular forms Math. Ann. 278 (1987) Tsuji, Takeshi $p$-adic étale coh …

It is true if $V$ is of dimension 1, essentially by class field theory (as you are considering only representations of $G_{\mathbb Q}$). Otherwise, it is still largely open for the following reasons. …