Terminology-history of p-adic representations

Question

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.

Answer 1

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.