I am reading the following article of [Berger, p8][1] 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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  [1]: https://mat.uab.cat/~masdeu/files/padichodge/berger_intro_theory_padic_reps.pdf