## $l$-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.

## Moduli spaces of arithmetic varieties with isomorphic $l$-adic cohomology

Given a positive integer $$d$$, a rational prime $$l$$ and a number field $$K$$, is it sensible to consider the moduli stack of $$d$$-dimensional varieties over $$K$$ whose $$l$$-adic cohomology rings are isomorphic as Galois modules?