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.