# $Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?

I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime distinct from the base field characteristic (the latter could be $0$). I would like to have a characterization of this vanishing in terms of etale cohomology of $l$-torsion sheaves.

Certainly, $Pic(X)\cong H^1_{et}(X,G_m)$; yet this only yields an injection of $Pic(X)/l$ into $H^1(X,\mu_l)$, and I don't know how to control the cokernel.

Upd. So, is there a general method that expresses $Pic(X)/l$ in terms of all of $H^i(X,\mu_{l^n})$ (for $i,n>0$)? What can be said here if $X$ is a variety over an algebraically closed field?

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$Pic(X)$ mod $l$ injects into $H^2(X,\mu_l)$ with cokernel the group of elements of order $l$ in the Brauer group of $X$. Depending on your $X$, the Brauer group may be known, or it may be as mysterious as a Tate-Shafarevich group. For example, for a complete smooth surface over a finite field, the Brauer group is conjectured to be finite, but this is not known (it's been proved to be equivalent to the Tate conjecture for the surface).
In characteristic zero, Hodge theory is the best way to approach this. Hodge theory would express $H^2(X,\mathbb \mu_l)=H^2(X,\mathbb Z/l)$ in terms of $H^2(X,\mathbb Z)$. You then use the fact that the classes in $H^2(X,\mathbb Z)$ that come from holomorphic line bundles are exactly Hodge classes, so you try to find the Hodge classes. This works best when $H^2(X,\mathbb Z)$ is nontorsion.
I think that in general $Pic(X)$ being $l$-divisible is rare, e.g. it is never such for any projective variety. One case where it is $l$-divisible is affine curves, where $H^2(\mu_l)$ is zero because $H^2$ of anything locally constant is zero.