$p$-adic periods

Question

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.

Answer 1

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.

Answer 2

Consider the elliptic curves defined over $\mathbb Q_p$. There are a couple of these for each $j$ invariant, and the $j$ invariant can be any element of $\mathbb Q_p$. Why should these periods do anything but fill up all of $\mathbb Q_p$?

Compare this to the case of complex periods, where only varieties defined over $\mathbb Q$ are considered. There are countably many of these, which immediately implies that not all complex numbers are periods.

It seems clear to me that, unless one is using a very strange definition of "period", one should only consider the periods of countably many varieties if one wants not all numbers to be periods. So you probably want to consider varieties over $\mathbb Q$, or perhaps algebraic subfields of $\mathbb Q_p$. This makes your de Rham cohomology countable, but it does nothing for your etale cohomology.

As S. Carnahan points out, you might want a version of etale cohomology with coefficients in a number field. But there is a big problem with this. If you have a version of etale cohomology for schemes over $\mathbb Q$ with coefficients in a number field, there is no reason to expect this cohomology theory to fail for schemes over $\mathbb F_l$. So one could get etale cohomology of $\mathbb F_l$-schemes with coefficients in a number field. One could then use a tensor product to change the coefficients to an $l$-adic field.

But an $l$-adic cohomology theory for schemes over $\mathbb F_l$ cannot possibly behave like etale cohomology. Instead, it must behave like crystalline cohomology, with the coefficient field extending as you extend the base field. This is because of an example of Serre.

So unless you come up with a version of etale cohomology that behaves horribly as you change the coefficients, or that magically changes its coefficient field as you change the base field, you are out of luck.

Answer 3

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.