$l$-adic periods?

Question

For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic periods with the associated Tannakian symmetry group has shown great success and promise recently - at least in genus 0 (mixed Tate motives) and genus 1 (mixed elliptic or modular motives) - with applications ranging from structure of the absolute Galois group (Deligne-Ihara conjecture) to transcendental number theory (multi-zeta values) to perturbative quantum field theory (Feynman amplitudes).

Betti and de Rham are only two of the realizations of motives. I wonder if there is a similar exploration of another comparison isomorphism, between de Rham and $l$-adic cohomologies. Is there a notion of "$l$-adic periods", concretely thought of as matrix coefficients of the isomorphism from de Rham to $l$-adic cohomology after some choice of bases? If so, what arithmetic significance and applications are they known to have?

Of course, there are other comparison isomorphisms as well, for which the question may well be trivial or uninteresting, just as it may be for de Rham/$l$-adic.

Answer 1

Yes. There are multiple isomorphisms between de Rham and $\ell$-adic cohomology. You can get one by combining the de Rham - Betti and Betti - $\ell$-adic isomorphisms. But this is silly as the periods you will get, in $\mathbb C\otimes_{\mathbb Q} \mathbb Q_\ell$, will just be the usual Betti periods.

A better approach is to use $\ell$-adic Hodge theory, which gives an isomorphism between de Rham cohomology and Betti cohomology when both are tensored to the field $B_{dR}$.

However, in this case, there is no need to use the motivic Tannakian group to study the periods. This is because of the identity $$H^*_{dR} (X, \mathbb Q) \otimes_{\mathbb Q} \mathbb Q_\ell = (H^*(X, \mathbb Q_\ell) \otimes B_{dR})^{ \operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )} $$ which implies that the periods in $B_{dR}$ are determined by the $\ell$-adic Galois representation restricted to $\operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )$ and thus are related to the Tannakian group of the category of $\operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )$-representations. This Tannakian group can be much smaller than the motivic Galois group.