Question

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?

Answer 1

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, 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.

Answer 2

There is the etale Abel-Jacobi map as defined by Nekovar. (See page 28 of the paper below)

http://www.math.jussieu.fr/~nekovar/pu/heegner.pdf

I have no idea - I don't know if it's conjectured, known, or known to be false - if this map provides a compatibility between the conjectures.