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Weil classes on every abelian sixfold of Weil type for Q(−3)\mathbb{Q}(\sqrt{-3})

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

Contributed by Benjamin Severn ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for K=Q(−3)K=\mathbb{Q}(\sqrt{-3}), of every Hermitian discriminant, are algebraic. The same holds for every abelian eightfold of Weil type for KK of signature (4,4)(4,4) with split discriminant, and every power of such a sixfold whose special Mumford--Tate group is SUHSU_H satisfies the Hodge conjecture. The proof applies the transfer theorem of the companion preprint to the Shephard--Todd group G34G_{34} with twenty reflections as local monodromies, and passes from eightfolds to sixfolds by a compensation argument. Two general tools are proved on the way. If every local monodromy of a transfer datum is conjugate to its inverse, the lattice condition of the transfer theorem follows from Brauer's induction theorem. Dominance of the period map follows from exact data at a maximal degeneration, by the nilpotent orbit theorem. We also give an explicit higher-dimensional form of Schoen's compensation argument, which shows that one decided cell of 2n2n-folds, n≥4n\ge4, decides every cell of sixfolds over the same imaginary quadratic field. No outside unrefereed manuscript is used. This version, with its computational certificates, is archived on Zenodo: https://doi.org/10.5281/zenodo.23422649

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). It began as part of the transfer-theorem manuscript, which went through four rounds of adversarial review by ChatGPT (OpenAI); after the split, this paper went through two further rounds on drafts v0.2 and v0.3 (verdicts: accept with revisions, then accept with minor revisions), and every objection raised was answered in the text or the certificates. This is version 0.5. The author set the research direction, chose which results to pursue, relayed the outside reviews and decided each revision. The boundary certificate for the G34 datum (the tuples, the lattice and dominance data, an independent SageMath reproduction, and a GAP chain check) consists of exact computations in GAP 4.14.0 and SageMath 10.7, supplied with READMEs and SHA-256 hashes of every file. The same version is archived on Zenodo, https://doi.org/10.5281/zenodo.23422649, with the certificates. The reviews were source-level: no human expert other than the author has checked the paper. No outside unrefereed manuscript is used.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
OpenAI
CodexVersion GPT-5.6 Sol
OpenAI
CodexVersion GPT-5.6 Luna
The GAP Group
GAPVersion 4.14.0
SageMath
SageMathVersion 10.7

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Version history

  1. v1Submitted by Benjamin SevernInitial depositCurrentOct 19, 2026