For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism
$$
H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}.
$$
If we pick $\mathbb{Q}$-bases for $H^i_{dR}(X)$ and $H^i_B(X,\mathbb{Q})$, the matrix entries of the comparison are complex numbers called periods of $X$, or more specifically periods of the motive $h^i(X)$.
Is there a comparison of cohomology theories that leads to a definition of 'period' for $p$-adic numbers, so that some but not all elements of $\mathbb{Q}_p$ are periods?
I have a specific example in mind: Riemann zeta values $\zeta(k)$, $k\geq 2$ an integer, are periods. An analogue in $\mathbb{Q}_p$ are the values $\zeta_p(k):=L_p(k,\omega_p^{1-k})$ of the $p$-adic $L$-function of Kubota-Leopoldt, where $\omega_p$ is the Teichmuller character. I'm hoping there's a sense in which $\zeta_p(k)$ is a period.
For $X$ a sufficiently nice variety over $\mathbb{Q}_p$, if I understand correctly there's a comparison isomorphism
$$
H^i_{dR}(X)\otimes B \to H^i_{et}(X,\mathbb{Q}_p)\otimes B,
$$
where $B$ is Fontaine's ring of $p$-adic periods. Elements of $B$ which show up as matrix coefficients with respect to $\mathbb{Q}_p$ bases of $H^i_{dR}(X)$ and $H^i_{et}(X,\mathbb{Q}_p)$ should also be called periods. Since both cohomologies in this comparison started out defined over $\mathbb{Q}_p$, every element of $\mathbb{Q}_p$ is a period in this sense.
Best Answer
You might want to look at Ogus' A p-adic analogue of the Chowla-Selberg Formula. There he defines p-adic periods for CM motives $X$ of rank 1 over a CM field $E$. Instead of using the dR-etale comparison isomorphism, he uses the semi-linear action of the crystalline Weil group $W$ (which is just the usual Weil group of $\mathbb{Q}_p$) on $H_{dR}(X/\bar K)$. Here, $K$ is any $p$-adic field over which the motive is realized. This action ultimately comes from the action of Frobenius on the crystalline cohomology of the special fiber which is identified with the de Rham cohomology of $X$ by results of Berthelot-Ogus. This identification is taking the role of the Betti-de Rham comparison isomorphism.
If one fixes a basis of $H_{dR}(X/\bar K)$, then each element $\sigma$ of $W$ determines a 1-by-1 matrix for the action of $\sigma$ with respect to the chosen basis (the coefficients are in $E \otimes \bar K$). So these (rank 1 CM) periods aren't single elements, but in fact are cocycles $W \to (E \otimes \bar K)^\times$, once we note that $W$ acts on the target through the right factor. Modulo coboundaries from $E \otimes \bar{\mathbb{Q}}$, the cocycle is independent of the choice of basis. Ogus proves that for Fermat curves, just as the complex periods are special values of the $\Gamma$-function, the $p$-adic periods can be computed in terms of special values of Morita's $p$-adic $\Gamma$-function.
But this notion of $p$-adic periods seems to generalize to any motive (not just CM, not just rank 1) with (potentially) good reduction at $p$. So one might hope for the values $\zeta_p(k)$ to appear as the values of a single "$p$-adic period cocycle" attached to some motive with good reduction over $\mathbb{Q}$. The number $k$ would correspond to an element $\sigma \in W$ which covers the $k$th power of the Frobenius automorphism of $\bar{\mathbb{F}}_p$. Any period cocycle coming from such a motive would vanish on inertia, so everything would be independent of the choice of $\sigma$ covering $k$. On the other hand, this would imply relations between the values $\zeta_p(k)$, so maybe it is too much to ask for the $\zeta_p(k)$ to come from the same cocycle.
Finally, I should add that Ogus' computation of p-adic periods on Fermat curves uses Faltings' de Rham-etale comparison isomorphism, so the latter is not irrelevant to this picture.