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

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

Contributed by Benjamin Severn ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Let K=Q(i)K=\mathbb{Q}(i). We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for KK whose Hermitian discriminant is the non-split class [−3][-3] are algebraic, and that every Hodge class on every power of such a sixfold is algebraic when its Hodge group is the full special unitary group. The proof starts from Galois covers of the projective line with monodromy group W(E8)⋅μ4W(E_8)\cdot\mu_4, branched over twelve points: the multiplicity space of the natural representation in the cohomology of such a cover is the first cohomology of an abelian sixfold of Weil type, and a certified rank computation at one explicit cover shows that these sixfolds fill an open set of the nine-dimensional moduli space. Algebraic cycles reach these sixfolds through Milne's theorem on Lefschetz classes, from two sources: Schoen's cycles on cyclic covers, which a computer search shows do not suffice here, and products of determinant lines of characters on abelian covers of curves. For the second source we prove a general theorem (Theorem C): if such a product is of Hodge type at every Galois conjugate, and each of the cyclic covers involved has at least two branch points and order dividing 1212 after removing the common factor of its branch exponents, then the product spans a space of algebraic classes. Its proof runs Schoen's symmetric-product construction on each factor, localises at a special fibre, transfers to a product of Fermat hypersurfaces, and applies Shioda's theorem on products of Fermat varieties. This version is archived on Zenodo: https://doi.org/10.5281/zenodo.23566607; its computational certificates: https://doi.org/10.5281/zenodo.23549514

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, SageMath and Python certificates, and drafted and revised the text. The main theorem's proof chain was checked by independent adversarial referees from a different model family (OpenAI GPT-6 Astra and GPT-5.6 Sol), whose repairs are incorporated; key lemmas, including the cone pairing, were reconstructed by two independent blind derivations before use. The paper then went through two rounds of cold reading by independent readers (Claude Opus and GPT-5.6 Sol) who saw only the paper, and every objection raised was answered in the text or the certificates. This is version 0.7. The author set the research direction, chose which results to pursue and decided each revision. The finite computations (the integer witness of the supplier identity and its rebuild from the group, the certified rank of the codifferential in ball arithmetic, the exact discriminant computation, the slot data, and Shioda's condition) were done in GAP 4.14.0, SageMath 10.7 and Python with python-flint and Normaliz 3.11, and are archived with a README and SHA-256 hashes at https://doi.org/10.5281/zenodo.23549514. The paper itself is archived at https://doi.org/10.5281/zenodo.23566607. The reviews were source-level: no human expert other than the author has checked the paper.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
OpenAI
CodexVersion GPT-5.6 Sol
The GAP Group
GAPVersion 4.14.0
SageMath
SageMathVersion 10.7
Normaliz
NormalizVersion 3.11

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Version history

  1. v1Submitted by Benjamin SevernInitial depositCurrentOct 22, 2026