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Algebraicity of Weil classes on abelian sixfolds of Weil type for Q(−3)\mathbb{Q}(\sqrt{-3}) with discriminant −2-2

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.GR — Group Theorymath.NT — Number Theory

Contributed by Benjamin Severn

Version 3 / Oct 14, 2026 / CC BY 4.0

Abstract

Let K=Q(−3)K=\mathbb{Q}(\sqrt{-3}). We prove that the Weil classes of every polarized abelian sixfold of Weil type for KK with Hermitian discriminant −2-2 are algebraic. This is a non-split discriminant, to which the theorems of Schoen and of Markman on split Weil classes do not apply. The proof uses Prym--Tyurin varieties of covers of P1\mathbb{P}^1 with monodromy the complex reflection group G32G_{32}, and combines known mechanisms in a new configuration. First, on every such variety the Weil classes of the six-dimensional isotypic factor are algebraic. They are obtained, by a Lefschetz-class correspondence supplied by Milne's theorem, from Schoen's algebraic Weil classes on fifty-one cyclic intermediate quotients; the choice of quotients is governed by an integral identity among characters of a torus, verified by an exact computation. Second, the ten-dimensional family of these covers dominates the nine-dimensional moduli space: we exhibit an explicit cover at which the period map has rank nine, by a certified computation. Third, a Hilbert-scheme and Baire-category argument spreads algebraicity from an open set to the whole moduli space. The discriminant of the family is identified exactly. All computations are supplied as reproducible certificates. For the members whose special Mumford--Tate group is the full special unitary group, an invariant-theoretic lemma then gives the Hodge conjecture for every power. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23377274

Provenance statement

This paper was produced in a research programme directed by the author, in which AI systems did most of the mathematical derivation, the computer-algebra code and the writing. Claude models (Anthropic) derived the arguments, wrote the GAP and SageMath certificates, and drafted and revised the text. Each step was checked by an independent adversarial referee from a different model family (OpenAI GPT-5.6 and GPT-6 models). The manuscript then went through five rounds of adversarial review by ChatGPT (OpenAI), on drafts v0.1 to v0.8, and every objection raised was answered in the text or the certificates. The author set the research direction, chose which results to pursue, relayed the outside reviews and decided each revision. All computations (certificates A-E: the lattice identity, the monodromy, the rank of the period map, the discriminant, and an independent check of the cup-product formula) are exact or interval-arithmetic computations in GAP 4.14.0 and SageMath 10.7 (FLINT/Arb). They were rerun from a clean copy, and are supplied with the source together with the SHA-256 hashes of their outputs. The reviews were source-level: no human expert other than the author has checked the paper. Remark 9.3 relies on a separate, unrefereed manuscript, as stated there. The same source is archived on Zenodo with the certificates and the review bundle: https://doi.org/10.5281/zenodo.23377274. Version 2 differs from version 1 only by this Zenodo link; the source file is unchanged.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
OpenAI
CodexVersion GPT-5.6 Sol
OpenAI
CodexVersion GPT-6 Astra
OpenAI
CodexVersion GPT-5.6 Luna
SageMath
SageMathVersion 10.7

References

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Show all 23 referencesShow fewer references
  1. E. Markman, Cycles on abelian 2n2n-folds of Weil type from secant sheaves on abelian nn-folds , arXiv:2502.03415.arXiv
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  7. Benjamin Severn, Computer certificates for: Algebraicity of Weil classes on abelian sixfolds of Weil type for $\Q(\sqrt{-3})$ with discriminant −2-2 , Zenodo, 2026, version 1.0, https://doi.org/10.5281/zenodo.23549518.DOI
  8. B. Severn, Weil classes on abelian sixfolds of Weil type for $\Q(i)$ with discriminant −3-3 , preprint, 2026.
  9. B. Severn, Weil classes on abelian sixfolds and eightfolds of Weil type for $\Q(\sqrt{-3})$ , preprint, 2026.
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  13. A. Weil, Abelian varieties and the Hodge ring , in: Collected Papers III, Springer, 1980, 421--429.

Version history

  1. v1Submitted by Benjamin SevernInitial depositSupersededOct 14, 2026
  2. v2Submitted by Benjamin SevernSubmitted as a new versionSupersededOct 19, 2026
  3. v3Submitted by Benjamin SevernSubmitted as a new versionCurrentOct 22, 2026