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Math-2589 · mathematics · Algebraic geometry

Tate classes on finite-field abelian varieties

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $A/\mathbb F_q$ be an abelian variety, let $r\geq0$, and let $\ell$ be a prime different from the characteristic. Is the cycle-class map \[\operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H_{\mathrm{et}}^{2r}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)}\] surjective? For algebraic cycles with rational coefficients on $A_{\overline{\mathbb F}_q}$, do numerical equivalence and $\ell$-adic homological equivalence coincide?

Short English statement

Let $A/\mathbb F_q$ be an abelian variety, let $r\geq0$, and let $\ell$ be a prime different from the characteristic. Is the cycle-class map \[\operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H_{\mathrm{et}}^{2r}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)}\] surjective? For algebraic cycles with rational coefficients on $A_{\overline{\mathbb F}_q}$, do numerical equivalence and $\ell$-adic homological equivalence coincide?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2589-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2589

Manuscript 1 (family 032, 2026-09-30): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Tate classes on finite-field abelian varieties

OpenAI's claim

The manuscripts claim algebraicity of rational Hodge classes on CM abelian varieties, with consequences for generalized Hodge classes and finite-field Tate classes. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
A complex abelian variety $A$ is CM if $\operatorname{End}(A)\otimes\mathbb Q$ contains a commutative semisimple algebra of dimension $2\dim A$. For every such $A$ and every $p\ge0$, OpenAI claims \[ \operatorname{CH}^p(A)_{\mathbb Q}\longrightarrow H^{2p}(A,\mathbb Q)\cap H^{p,p}(A) \quad\text{is surjective}. \] This also applies to arbitrary finite products of CM abelian varieties, including their powers. If $W\subset H^k(A,\mathbb Q)$ is a rational Hodge substructure and $W(r)$ is effective (all Hodge bidegrees are nonnegative), it claims a closed algebraic $Z\subset A$ of codimension at least $r$ with $W\subset\ker[H^k(A,\mathbb Q)\to H^k(A\setminus Z,\mathbb Q)]$. The manuscript additionally asserts that, for every abelian variety $A/\mathbb F_q$, every $r\ge0$ and prime $\ell\ne\operatorname{char}\mathbb F_q$, \[ \operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H^{2r}_{\mathrm{\acute et}}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)} \quad\text{is surjective}. \] Numerical and $\ell$-adic homological equivalence are claimed to agree on rational cycles. Over any algebraically closed field, for an abelian $g$-fold and rational ample class $h$, the form $(-1)^r\deg(xyh^{g-2r})$ is claimed positive definite on the real extension of $\ker[h^{g-2r+1}:A^r_{\mathrm{num}}\to A^{g-r+1}_{\mathrm{num}}]$ for $0\le r\le g/2$.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): The argument realizes four-factor balanced tensors using unitary theta constructions. Elementary switches between equal-sum lists, followed by algebraic polarization contractions and descent of coefficients, build arbitrary balanced CM tensors from these elementary cycle classes.

Formal verification

Family 032 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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