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Math-2192 · mathematics · Algebraic geometry

The Tate Conjecture

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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.
Let $X$ be smooth and projective over a finite field, let $r\geq0$, and let $\ell$ differ from the field characteristic. Must every Frobenius-invariant class in $H^{2r}_{\mathrm{et}}(X_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))$ be a $\mathbb Q_\ell$-linear combination of cycle classes of codimension-$r$ algebraic cycles defined over the finite field? Applying the assertion after finite field extensions gives the usual Tate-class formulation.

Formulation references: [S1], [S2].

Short English statement

Let $X$ be smooth and projective over a finite field, let $r\geq0$, and let $\ell$ differ from the field characteristic. Must every Frobenius-invariant class in $H^{2r}_{\mathrm{et}}(X_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))$ be a $\mathbb Q_\ell$-linear combination of cycle classes of codimension-$r$ algebraic cycles defined over the finite field? Applying the assertion after finite field extensions gives the usual Tate-class formulation.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2192-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2192

The Tate corollary supplies the required cycle representatives for abelian varieties over finite fields, in all codimensions. This page asks the same assertion for arbitrary smooth projective varieties over finite fields, including varieties outside the abelian class. The result therefore covers a restricted class of the recorded question.

The Tate Conjecture

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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