$l$-adic periods?

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 Feynman diagrams.

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 are they known to have, similar to $ \pi$ or $ \zeta(2)$ ?

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.