Let Y be a smooth projective variety over a global field, and let i∈Z. Write operatornameCH^i(Y)_0 for the kernel of the cycle class map on operatornameCH^i(Y). Let L(H^2i-1(overlineY,Q_ell),s) be the associated L-function. Beilinson–Bloch conjecture. dim_QoperatornameCH^i(Y)_0=operatornameord_s=iL(H^2i-1(overlineY,Q_ell),s). This predicts that the rank of the homologically trivial codimension-i Chow group equals the order of vanishing of the relevant L-function. The source recalls it as the conjectural description of the kernel of the cycle class map over a global field; no resolution status is supplied.

Beilinson–Bloch conjecture for smooth projective varieties over global fields

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Let $Y$ be a smooth projective variety over a global field, and let $i\in\mathbb{Z}$ . Write $\operatorname{CH}^i(Y)_0$ for the kernel of the cycle class map on $\operatorname{CH}^i(Y)$ . Let $L(H^{2i-1}(\overline{Y},\mathbb{Q}_{\ell}),s)$ be the associated $L$ -function. Beilinson–Bloch conjecture.

\dim_{\mathbb{Q}}\operatorname{CH}^i(Y)_0=\operatorname*{ord}_{s=i}L(H^{2i-1}(\overline{Y},\mathbb{Q}_{\ell}),s).

This predicts that the rank of the homologically trivial codimension- $i$ Chow group equals the order of vanishing of the relevant $L$ -function. The source recalls it as the conjectural description of the kernel of the cycle class map over a global field; no resolution status is supplied.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Beilinson–Bloch conjecture for smooth projective varieties over global fields
Let $X$ be a smooth projective variety over a global field $k$ . For $0\leq i\leq\dim X$ , let $\operatorname{CH}^i(X)_0$ denote the subgroup of codimension- $i$ cycles that are homologically trivial, and let $L(H^{2i-1}(X_{\overline{k}},{\mathbb Q}_\ell),s)$ be the associated Hasse–Weil $L$ -function.

Beilinson–Bloch conjecture. For every $0\leq i\leq\dim X$ ,
$\dim_{\mathbb Q}\operatorname{CH}^i(X)_0=\operatorname*{ord}_{s=i}L(H^{2i-1}(X_{\overline{k}},{\mathbb Q}_\ell),s).$
This relates the rank of the group of homologically trivial cycles to the order of vanishing of the relevant $L$ -function. The supplied text gives no resolution status, so the conjecture is recorded as open.
source: Matt Broe, “The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields”, arXiv:2509.03602 (2026).

References

Primary source

Matt Broe, “On the Beilinson-Bloch conjecture over function fields”, arXiv:2505.00696 (2026).

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