In his paper Fonctions L p-adiques, Pierre Colmez says:
Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques des variétés algébriques ne pouvaient pas vivre dans $\mathbf C_p$.
(Tate showed that there does not exist in $\mathbf C_p$ a $p$-adic analogue of $2i\pi$, and therefore that the $p$-adic periods of algebraic varieties cannot live in $\mathbf C_p$.)
He further explains that this realization led Fontaine to construct his complicated 'rings of periods'. He gives, as reference for the quoted claim, Tate's paper p-divisible groups. However, nothing that I've read in Tate's paper seems to immediately justify this surprising claim. I've read a bit about Fontaine's rings and I have an idea of the role they play in studying $p$-adic representations, but I'm not quite sure how to formally express "that there does not exist in $\mathbf C_p$ a $p$-adic analogue of $2i\pi$".
My feeling is that he is thinking about periods on curves, where the formal residue theorem of Serre-Tate allows us to explicitly describe Riemann-Roch-Poincaré duality in terms of a formal integration pairing. Over $\mathbf C$, this pairing can be calculated analytically by Cauchy's Residue Theorem, but it requires "division by $2i \pi$"... and somehow, we can't mimick this inside $\mathbf C_p$? is this right? if so, why can't we mimick it?
I would welcome any clarification or explanation of this matter. Thanks!
Edit: This user comment on the blog post linked to by anon, seems to suggest that my hunch is correct. So how can we justify this claim, and get a good feel for the necessity of Fontaine's rings? And, how is this related to Tate's paper?