The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an $\ell$-adic Abel-Jacobi map
$\phi$\phi : Ch^j(V)_0 \to H^1(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell})(j))$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 : 2 added 2 characters in body The Bloch-Kato conjecture is actually more precise than that. As mentioned by Hunter Brooks, there is indeed an$\ell$-adic Abel-Jacobi map$\phi : Ch^j(V) Ch^j(V)_0 \to H^1(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell})(j))$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 : 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$\phi : Ch^j(V) \to H^1(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell})(j))$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 :