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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.

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Hodge-Tate appears (in that name) at least as early as Serre's book "Abelian $\ell$-adic representations" (1968). Someone with a better knowledge of the literature may be able to beat that. – tkr Oct 25 '11 at 21:45
up vote 6 down vote accepted

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.

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Just to clarify -- although the concept of a Hodge-Tate representation is clearly (implicitly at least) in Tate's 1967 article (which also contains at least one deep theorem about them, namely that the Tate module of an ab var is H-T), the actual adjective "Hodge-Tate" is not explicitly mentioned there -- perhaps it was Serre who introduced the terminology in 1968. And no surprise really -- I have seen Tate, in a talk, considering "the module T:=projlim_n E[p^n] analysed by Weil", presumably because he was too modest to call it "the Tate module". – Kevin Buzzard Oct 26 '11 at 6:44

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