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Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over Q(−3)\mathbb{Q}(\sqrt{-3}) with Hermitian determinant −2-2

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometrymath.GR — Group Theorymath.NT — Number Theory

Contributed by Antti Hippeläinen

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

Let K=Q(−3)K=\mathbb{Q}(\sqrt{-3}). We give a proof that every Weil class is algebraic on every complex abelian sixfold of Weil type for KK whose Hermitian form has signature (3,3)(3,3) and determinant −2-2 modulo norms. These forms are not split, so neither Schoen's theorem on sixfolds nor the theorem on sixfolds claimed in a preprint of Markman, both of which concern split forms, applies. We model the sixfolds by the part of the Jacobian of a cover C→P1C\to\mathbb{P}^1, branched at thirteen points, on which its group, the complex reflection group G32G_{32}, acts through its reflection representation. A degeneration argument with a ball-arithmetic certificate shows that this family dominates the nine-dimensional Shimura variety. On every member, an integral relation among characters induced from linear characters of subgroups, realized by correspondences with coefficients in the group algebra, reduces the Weil classes to the determinants of eigenspaces of eleven cyclic subcovers. A theorem of Schoen makes these determinants algebraic for the three étale subcovers; for the eight ramified ones we prove it for all cyclic covers of curves of degree at least 33 with at least 33 branch points and fully paired branching data, through a cone over a Fermat variety and Shioda's results. Specialization and isogenies then reach every sixfold. The determinant of the Hermitian form and the relation among characters are exact computations.

Provenance statement

The proofs, the computations and the text of this note were produced by large language models (Claude models of Anthropic running as autonomous agents) in October 2026, and checked by other instances of the same models acting as referees, which recomputed the numbers. The author directed the work, read through the note and checked that all references are valid. However, no human expert has checked the proofs. The computations of Appendix A are reproduced by the programs at https://github.com/ahippelainen/weil-nonsplit-abelian-sixfolds-public.

Tools used

Anthropic
Claude OpusVersion 5.5
GAP-system
GAPVersion 4.16.1
Python Software Foundation
Python with python-flint 0.9.0

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Version history

  1. v1Submitted by Antti HippeläinenInitial depositSupersededOct 18, 2026
  2. v2Submitted by Antti HippeläinenSubmitted as a new versionCurrentOct 28, 2026