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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 $H^1_f(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell}))(j))$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

$\phi_j : \mathrm{Ch}^j(V)_{0} \to H^{1}_{f}(G_K,H^{2j-1}_{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell}))(j))$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?
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Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V){0}$ \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 $\mathrm{Ch}^j(V){0} \mathrm{Ch}^j(V)_{0} \otimes \mathbf{Q}$ as a $\mathbf{Q}$-vector space is given by the order of vanishing of the L-function $L(s,H^{2j-1}{\mathrm{et}}(V L(s,H^{2j-1}_{\mathrm{et}}(V \times{K} times_{K} \overline{K},\mathbf{Q}_{\ell}))$ 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 $H^1_f(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell}))(j))$. 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

$\phi_j : \mathrm{Ch}^j(V){0} mathrm{Ch}^j(V)_{0} \to H^{1}{f}(G_K,H^{2j-1}{\mathrm{et}}(V\times{K}\overline{K},\mathbf{Q}_{\ell}))(j))$ 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?
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Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^j(V)\mathrm{Ch}^{j}(V){0}$ 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 $L(s,H^{2j-1}{\mathrm{et}}(V\times\mathrm{et}}(V \times{K}\overline{K},\mathbf{Q}_{\ell}))$ 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

$\phi_j : \mathrm{Ch}^j(V){0} \to H^{2j-1H^{1}{\mathrm{et}}(V\times_{K}\overline{K},\mathbf{Q}_{\ell}))(j)$ 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?
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