Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Theorem · version 2
Tate is proved for abelian fourfolds over finite fields
Statement
For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.
Assumptions
- A is an abelian variety of dimension four over a finite field.
- ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group.
Scope
A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.
Evidence
Source reports
Connections
In this problem · publication v14 · Selected context
Hodge conjecture →
Different statement · related variant · Proved
Analogue
Hodge asks for Betti Hodge classes over C; Tate asks for Galois-invariant ℓ-adic classes over finite fields. Broe’s proof uses Markman’s geometric input with further specialization arguments.
This relationship is not a direct implication from the fourfold corollary alone, or a new case of canonical Hodge.
Details & sources
Abelian fourfolds · Markman (2025) →
Proved · version 2
The rational Hodge conjecture holds for every complex abelian variety of dimension four.
- A is a complex abelian variety.
- Complex dimension of A is 4.
- Rational coefficients in cohomology and cycles.
Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.
Tate is proved for abelian fourfolds over finite fields →
Proved · version 2
For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.
- A is an abelian variety of dimension four over a finite field.
- ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group.
Scope: A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.
Relationship not independently reviewed.
Connection in Hodge conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v15
Hodge conjecture →
Different statement · related variant · Proved
Analogue
Hodge asks for Betti Hodge classes over C; Tate asks for Galois-invariant ℓ-adic classes over finite fields. Broe’s proof uses Markman’s geometric input with further specialization arguments.
This relationship is not a direct implication from the fourfold corollary alone, or a new case of canonical Hodge.
Details & sources
Abelian fourfolds · Markman (2025) →
Proved · version 2
The rational Hodge conjecture holds for every complex abelian variety of dimension four.
- A is a complex abelian variety.
- Complex dimension of A is 4.
- Rational coefficients in cohomology and cycles.
Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.
Tate is proved for abelian fourfolds over finite fields →
Proved · version 2
For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.
- A is an abelian variety of dimension four over a finite field.
- ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group.
Scope: A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.
Relationship not independently reviewed.
Connection in Hodge conjecture →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Broe · Tate for abelian fourfolds over finite fields →
- Theorem 1.1; introductory deduction of all degrees, PDF pages 1–2 → · Original passage ↗
Recorded wording
For an abelian fourfold over F_q and ℓ ≠ char(F_q), the ℓ-adic cycle-class map in codimension two surjects onto Galois-invariant H⁴; other codimensions follow from divisor and Lefschetz results.
paraphrase · Only metadata and original OpenMath paraphrases are published; full text stays upstream.
Matt Broe · Preprint · Publication date Aug 19, 2026
Source-reported manuscript at this exact revision; independent mathematical review is pending.
Broe · Tate for abelian fourfolds over finite fields — manuscript ↗Manuscript / exposition · external
Revision & attribution
Source revision: arxiv-v1. Observed Oct 10, 2026.
The cited source does not report AI involvement; no absence claim is inferred.
Broe · Tate for abelian fourfolds over finite fields — manuscript: arxiv-v1. External manuscript. No OpenMath proof execution, formalization or independent mathematical acceptance is inferred.
Markman · cycles from secant sheaves →
- Corollary 1.6.1, PDF page 9; Theorem 1.5.1 → · Original passage ↗
Recorded wording
The rational Hodge conjecture holds for every complex abelian variety of dimension four.
paraphrase · No upstream full text or code is republished by this record.
Eyal Markman · Preprint
Author preprint at the cited revision; journal acceptance has not been established by this curation.
Revision & attribution
Source revision: arxiv-v2. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Jun 8, 2025.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-hodge-broe-tate-fourfolds
- Statement version
- 2
- SHA-256
- 6efd36122b95aeaa55e8bd17704097bdb2f43b986a345604f4b61f9146e4cd39