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

Question

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!

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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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Answer 1

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.