I am reading the following article of Berger, p8 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

this follows from the result that (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!

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.