Where appears for the first time the term Hodge-Tate representation.
Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.



Terminology-history of p-adic representations
The notion of Hodge-Tate decomposition has been introduced by Tate, in 1967. 
(The paper itself is called *$p$-divisible groups*, and it appeared
in the Proceedings of a conference on local fields that took place in Driebergen.)
There, he shows that over a $p$-adic field, the $p$-adic Tate module $T\_p(G)$ of an Abelian variety with good reduction $G$ possesses a kind of Hodge decomposition:
$$ T\_p(G) \otimes\_{\mathbf Z\_p} {\mathbf C\_p} \simeq \mathbf C\_p^g \oplus \mathbf C\_p(-1)^g,$$
as Galois modules, $\mathbf C\_p(-1)$ denoting the action via the cyclotomic character,
and $g$ being the dimension of $G$.
This is reminiscent of the Hodge decomposition over the complex numbers.


It has been proved later, by Faltings, that the $p$-adic étale cohomology of any 
smooth projective variety over a $p$-adic local field admits a similar decomposition
when tensored with $\mathbf C\_p$:
$$ H^n(X,\mathbf Z\_p)\otimes\mathbf C\_p \simeq \bigoplus \mathbf C\_p(i)^{h^{i,n-i}},$$
where $h^{i,n-i}=\dim H^i(X,\Omega\_X^{n-i})$ are the Hodge numbers.


Now, there are other cohomology theories, the crystalline, the De Rham, etc.
and the rings forged by Fontaine play the rôle that $\mathbf C\_p$ (technically,
the direct sum of all $\mathbf C\_p(i)$) plays for the Hodge cohomology.


