TITLE: $l$-dependence of the group of homologically zero cycles
QUESTION [4 upvotes]: Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper he mentiones that the kernel $CH^i(X)_l^0$ might depend on $l$.
What is known about this issue? Is there an example of $X$ and two primes $l_1,l_2$ such that $CH^i(X)_{l_1}^0\neq CH^i(X)_{l_2}^0$?

REPLY [5 votes]: the way it's stated, with integral cohomology, it certainly can depend on $\ell$. This is because of the occurrence of torsion classes Clearly a $p$-torsion class is sent to $0$ by this map for $\ell\neq p$, so it is sufficient to find an $\ell$-torsion class that is not sent to zero under this map.
But the Kummer exact sequence shows that for $i=1$, the kernel of the map is $\ell$-divisible. Hence any non-divisible torsion class, like a torsion point on an elliptic curve over a number field, or the canonical bundle of an Enriques surface, will do.
However with $\mathbb Q_\ell$ it may still be independent of $\ell$, depending on the base field. For algebraically closed fields it should be independent, because of the standard conjecture D which says the kernel would just be the group of cycles numerically equivalent to $0$. For arbitrary fields there are counteredamples as Mikhail Bondarko pointed out. For finitely generated fields I'm not sure.
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