# Why does $\mathbb C_p$ not contain the periods?

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

(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.

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.