TITLE: $p$-adic periods
QUESTION [18 upvotes]: 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.

REPLY [2 votes]: There is a notion of $p$-adic period coming from the Frobenius action on $p$-adic cohomology. Suppose $X$ is a smooth variety over $\mathbb{Q}$, and let $p$ be a prime. If there is a smooth model $\tilde{X}$ of $X$ over $\mathbb{Z}_{(p)}$, then $H_{dR}^*(X)\otimes\mathbb{Q}_p$ depends only on the special fiber $\tilde{X}_p:=\tilde{X}\times \mathrm{Spec}\,\mathbb{F}_p$. There are several different cohomology theories to explain this: Monsky-Washnitzer, crystalline, rigid.
In particular, the Frobenius endomorphism of $\tilde{X}_p$ induces a linear map
$$
F_p:H_{dR}^*(X)\otimes\mathbb{Q}_p\to H_{dR}^*(X)\otimes\mathbb{Q}_p.
$$
If we choose a basis for $H^*_{dR}(X)$, we may represent $F_p$ by a square matrix with entries in $\mathbb{Q}_p$, and $\mathbb{Q}$-linear combinations of matrix entries might be viewed as $p$-adic periods of $X$. The $p$-adic zeta values $\zeta_p(k)$ (and more generally $p$-adic multiple zeta values) are $p$-adic periods in this sense.
There is a discussion of $p$-adic periods at the end of Section 5.4 of Francis Brown's Notes on motivic periods. Another reference (given by Nils Matthes in the comments) is Go Yamashita's Bounds for the dimensions of $p$-adic multiple $L$-value spaces, in particular Remark 3.9.