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Math-2267 · mathematics · Number theory

A classwise CM implication and a one-product coniveau certificate

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
In a setting where the Hodge conjecture implies the Tate conjecture, must the generalized Hodge conjecture also imply the generalized Tate conjecture? In particular, does this implication hold for CM abelian varieties? The generalized assertions concern coniveau, not only individual cycle classes.

Formulation references: [S1], [S2].

Short English statement

In a setting where the Hodge conjecture implies the Tate conjecture, must the generalized Hodge conjecture also imply the generalized Tate conjecture? In particular, does this implication hold for CM abelian varieties? The generalized assertions concern coniveau, not only individual cycle classes.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2267-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2267

The CM result supplies algebraic supports for the generalized Hodge assertion, and its finite-field corollary represents ordinary Tate classes by cycles. The question asks whether a generalized Tate statement follows from generalized Hodge, with the relevant coniveau condition. No such generalized Tate implication is asserted in the selected paper. These conclusions are related to its premises, without proving a branch of the requested implication.

A classwise CM implication and a one-product coniveau certificate

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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