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Math-2266 · mathematics · Number theory

A new-Lefschetz defect reduction for rational Hodge pairings

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.
Let $A$ be a CM abelian variety over $\overline{\mathbb Q}$ with good reduction $A_0$ in characteristic $p$, and let $c$ be an absolute Hodge class on $A$. For a complementary-dimensional Lefschetz class $x$ on $A_0$, the specialization pairing $x\cdot\overline c$ lies in $\mathbb Q_\ell$ for every $\ell\neq p$. Must it be rational and independent of $\ell$? The question also asks for the corresponding property for arbitrary abelian varieties with good reduction.

Formulation references: [S1], [S2].

Short English statement

Let $A$ be a CM abelian variety over $\overline{\mathbb Q}$ with good reduction $A_0$ in characteristic $p$, and let $c$ be an absolute Hodge class on $A$. For a complementary-dimensional Lefschetz class $x$ on $A_0$, the specialization pairing $x\cdot\overline c$ lies in $\mathbb Q_\ell$ for every $\ell\neq p$. Must it be rational and independent of $\ell$? The question also asks for the corresponding property for arbitrary abelian varieties with good reduction.

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2266-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2266

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting results are manuscript 1 (family 001; 2026-09-23), manuscript 2 (family 032; 2026-09-30). The manuscript also gives additional strength beyond the requested conclusion; that excess is not required for this problem's solved-claim label. Other linked papers contribute only their stated special cases; those weaker results alone would not resolve the complete question.

A new-Lefschetz defect reduction for rational Hodge pairings

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 001; 2026-09-23), the asserted result is that reduction of rational Hodge classes on abelian varieties preserves rational intersection numbers with divisor products, in crystalline as well as prime-to-characteristic cohomology, including characteristic two; In manuscript 2 (family 032; 2026-09-30), the asserted result is that rational Hodge classes on CM abelian varieties in all dimensions are algebraic, with the stated Tate and Hodge-standard consequences; In manuscript 3 (family 032; 2026-09-30), the asserted result is that the rational Hodge assertion holds on every self-power of the stated hyperbolic Weil sixfolds of signature $(3,3)$ and on every self-power of imaginary-quadratic-action varieties of dimension at most five.

Mathematical statement of the claimed result

Manuscript 1.
Let an abelian variety $A/\overline{\mathbb Q}$ of dimension $d$ have good reduction $A_0/\overline{\mathbb F}_p$ at a place $w$. For any rational Hodge class $\gamma\in H^{2r}(A(\mathbb C),\mathbb Q(r))$, $0\le r\le d$, and Cartier divisors $D_1,\ldots,D_{d-r}$ on $A_0$, the claim is existence of one rational number $q$ with \[ \operatorname{Tr}_{\ell}(\gamma_{0,\ell}\prod_i c_{1,\ell}(D_i))=q\quad(\ell\ne p),\qquad \operatorname{Tr}_p(\gamma_{0,p}\prod_i c_{1,\rm cris}(D_i))=q. \] The first specialization is smooth-proper étale specialization; the second is de Rham--crystalline specialization in the completed $p$-adic comparison field. The trace normalization is in the highest Tate twist, with the trace of a zero-cycle equal to its degree. For $r=d$ the divisor product is one. This is a common rational intersection-number assertion, not itself a full algebraicity assertion for every Hodge class.
Manuscript 2.
If $A$ is any complex abelian variety of CM type, then for every integer $p\ge0$ the claim is \[ \operatorname{cl}_B:\operatorname{CH}^p(A)\otimes_{\mathbb Z}\mathbb Q \twoheadrightarrow H^{2p}(A,\mathbb Q)\cap H^{p,p}(A). \] The intersection is inside complex cohomology, so this asserts algebraicity of every rational Hodge class, rather than merely classes with extra symmetry. A CM abelian variety has a commutative semisimple algebra of rational endomorphisms of dimension $2\dim A$. Finite products and all powers of CM abelian varieties are included in the assertion.
Manuscript 3.
Take a complex abelian variety $A$, with $\dim A\le5$, and an imaginary quadratic field embedded in $\operatorname{End}(A)\otimes\mathbb Q$. For every positive integer $m$ and every codimension $p\ge0$, the claim is \[ \operatorname{CH}^p(A^m)\otimes_{\mathbb Z}\mathbb Q \twoheadrightarrow H^{2p}(A^m,\mathbb Q)\cap H^{p,p}(A^m) \] under the Betti cycle-class map, with the intersection inside complex cohomology. It includes all rational Hodge classes on every power, rather than only the distinguished Weil classes.

Outline of the argument

For manuscript 1 (family 001; 2026-09-23), the construction moves Hodge tensors through integral models of Hodge-type Shimura varieties while retaining a chosen divisor. An endomorphism-compatible filtration and balanced Weil classes reduce the comparison to CM varieties, where algebraic cycles supply the common rational value.

For manuscript 2 (family 032; 2026-09-30), balanced CM tensors are reduced to four-factor switch relations. Theta one-forms on compact ball quotients supply algebraic representatives for those relations, and polarization contractions assemble the general tensor before rational coefficient descent.

For manuscript 3 (family 032; 2026-09-30), a tensor criterion reduces algebraic supply to determinant classes at the given Hodge group. A geometric seed supplies the Weil plane, twisted-sheaf deformation transports it locally, and Hilbert spaces with a Baire argument extend algebraicity to all marked periods.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 001; 2026-09-23), manuscript 2 (family 032; 2026-09-30), manuscript 3 (family 032; 2026-09-30).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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