I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $\mathbb G_m$:

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I know the example for the infinite place: let $w=\frac{dz}{z}$ be the specific holomorphic differential on $\mathbb G_m(\mathbb C)=\mathbb C^{\times}$, $S^1$ be the unit circle in $\mathbb C^{\times}$ which is a generator for $H_1(\mathbb C^{\times}, \mathbb Z)$, then the period is the intergal:

$\int_{S^1} w= \int_{S^1} \frac{dz}{z}=2 \pi i.$

which is clear from standard complex analysis. My question is how to compute the $p$-adic period as the comparison theorem in $p$-adic Hodge theory he used:

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I think this is a standard exercise in $p$-adic Hodge theory, but I couldn't find some references about the computation $|t_p|_p=p^{-\frac{1}{p-1}}$ (here the p-adic valuation is normalized such that $|p|_p=1/p$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.

Thank you for any help.


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