Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)\_{0}$
be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture of Beilinson and Bloch predicts that the dimension of $\mathrm{Ch}^j(V)\_{0} \otimes \mathbf{Q}$ as a $\mathbf{Q}$-vector space is given by the order of vanishing of the L-function $L(s,H^{2j-1}\_{\mathrm{et}}(V \times\_{K} \overline{K},\mathbf{Q}\_{\ell}))$ at its central critical point. On the other hand, the Bloch-Kato conjecture predicts that this order of vanishing is equal to the dimension of the Bloch-Kato Selmer group $H^1\_f(G\_K,H^{2j-1}\_{\mathrm{et}}(V\times\_{K}\overline{K},\mathbf{Q}\_{\ell})(j))$. So it seems reasonable to ask: is there a natural map 


$\phi\_j : \mathrm{Ch}^j(V)\_{0} \to H^{1}\_{f}(G\_K,H^{2j-1}\_{\mathrm{et}}(V\times\_{K}\overline{K},\mathbf{Q}\_{\ell})(j))$ `


which is "close" to being an isomorphism, which explains the compatibility of these conjectures?



Compatibility of Bloch-Kato and Beilinson-Bloch
The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map


$$\phi : Ch^j(V)\_0 \to H^1(G\_K,H^{2j-1}\_{\mathrm{et}}(V\times\_{K}\overline{K},\mathbf{Q}\_{\ell})(j))$$


The map $\phi$ is also called the cycle class map and is defined for any field $K$, say of characteristic $0$.


Now if $K$ is a number field, the conjecture of Bloch-Kato predicts that


(1) $\phi$ takes values in $H^{1}\_f(G\_K,\cdot)$.


(2) $\phi \otimes \mathbf{Q}\_\ell$ is an isomorphism.


In fact (1) is a purely local question : this is really a question about the Abel-Jacobi map associated to a variety over $\mathbf{Q}\_p$ (where $p$ can be equal to $\ell$ or not). The conjecture (2) together with the Beilinson-Bloch conjecture implies the statement you mention about the order of vanishing.


You can find a good survey on the $\ell$-adic Abel-Jacobi map and more details about what is known here :


J. Nekovar, [p-adic Abel-Jacobi maps and p-adic heights](http://people.math.jussieu.fr/~nekovar/pu/banff.pdf), In: The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), 367 - 379, CRM Proc. and Lect. Notes 24, Amer. Math. Soc., Providence, RI, 2000.


See also the "Conjecture $\mathrm{Mot}\_\ell$" in the following article :


M. Flach, [The Equivariant Tamagawa Number Conjecture : A Survey](http://www.math.caltech.edu/papers/baltimore-final.pdf).


