I am reading the following article of [Berger, p8](https://mat.uab.cat/%7Emasdeu/files/padichodge/berger_intro_theory_padic_reps.pdf) and I don't understand the idea:



> 
> $C\_p:=\widehat{\overline{\mathbb Q\_p}}$ does not contain the periods
> 
> 
> 


The text seem to reason as follows



> 
> (under some conditions)
> $$ H^0(K, C\_p(\chi^{-1})) = \{x \in C\_p \, : \, gx = \chi(\sigma)x \forall \sigma \in G\_K\} =0 $$
> for some character $\chi:G\_K \rightarrow \mathbb Z\_p^\times$, where this set is the "set of periods".
> 
> 
> 


Question:


* How does this relate to the classical of notion of periods/why is this set periods?


Details / explanations would be appreciated!




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My thoughts: (Can ignore)


What I know: one formualtion of periods in the $\mathbb Q/\mathbb C$ setting is that that they are coefficients in the comparison iso.
$$ C\_{dR}: H^n\_{dR}(X(\mathbb C), \mathbb Q) \otimes \mathbb C \simeq H^n\_{Betti}(X(\mathbb C), \mathbb Q) \otimes\_{\mathbb Q} \mathbb C $$
I believe our example here is consider $\mathbb G\_{m,\mathbb Q\_p}$.


What I don't see: how Galois groups even come into play.




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Why does $\mathbb C_p$ not contain the periods?
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 $2\pi i\mathbb Q$. The comparison theorem you recalled works because after extension of scalars to $\mathbb C$, you can multiply by $2\pi i$.


However, the observation of Tate recalled by Laurent Berger that you mentioned above tells you that $\mathbb C\_{p}(1)=\{0\}$. Thus $\mathbb C\_{p}(1)$, which is the étale realization of $\mathbb Q(1)$ after extension of scalars to $\mathbb C\_{p}$ is not isomorphic to $\mathbb C\_{p}$, which is the de Rham realization of $\mathbb Q(1)$ after extension of scalars to $\mathbb C\_{p}$. You would encounter similar problems if you were to believe that the period ring for the Betti-de Rham comparison isomorphism is $\mathbb R$ (in that case there would be no compatibility in the filtration and Hodge structure)


So one needs to extend to a ring which contains something like $2\pi i$ in the $p$-adic world, and that is the ring $B\_{{dR}}$ of Jean-Marc Fontaine.


