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A transfer theorem for Weil classes on Prym--Tyurin factors of Galois covers

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

Let KK be an imaginary quadratic field, GG a finite group, σ\sigma an irreducible representation of GG with character field KK, and SS a connected component of a Hurwitz space of GG-covers of P1\mathbb{P}^1 with fixed local monodromy. When the σ\sigma-isotypic factors of the Jacobians are abelian 2n2n-folds of Weil type for KK, we prove that the Weil classes of every polarized abelian 2n2n-fold of Weil type for KK with Hermitian discriminant d\mathfrak{d} are algebraic, provided three conditions hold: a lattice condition (L) on the characters of a group of multiplicative type attached to the cover, dominance (D) of the period map of SS, and the equality of the Hermitian discriminant of the family over SS with d\mathfrak{d}. None of the conditions refers to a particular group. The Hodge conjecture for every power of a member AA follows when the special Mumford--Tate group of AA itself is SUHSU_H; a statement about the generic member of the family does not suffice, and members with smaller Hodge group are not treated here. Positive instances of the conditions are verified by finite input certificates, so a new example needs new certificates, whose validation may need its own arguments, and no new transfer proof. The theorem recovers the case G=G32G=G_{32}, K=Q(−3)K=\mathbb{Q}(\sqrt{-3}), d=−2\mathfrak{d}=-2 of the author's earlier preprint (hexagon:2610.00246), with every step of that proof an instance of a general lemma. A second source of cycles, balanced combinations of several characters of one subgroup, is available conditionally on a theorem of an unrefereed manuscript on abelian covers. With it the datum (Q(i),[−3])(\mathbb{Q}(i),[-3]) carried by W(E8)⋅μ4W(E_8)\cdot\mu_4 satisfies every hypothesis, so, conditionally on that theorem, the Weil classes of every abelian sixfold of Weil type for Q(i)\mathbb{Q}(i) with discriminant −3-3 are algebraic. For this datum the Schoen-only condition (L) fails. A companion preprint applies the transfer theorem, with new tools, to prove that the Weil classes of every abelian sixfold of Weil type for Q(−3)\mathbb{Q}(\sqrt{-3}), of every Hermitian discriminant, are algebraic. We also state what the theorem cannot reach. This version, with its computational certificates, is archived on Zenodo: https://doi.org/10.5281/zenodo.23422651

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 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 four rounds of adversarial review by ChatGPT (OpenAI), on drafts v0.6 to v0.12 (verdicts: accept with revisions, accept with minor revisions twice, accept with revisions), and every objection raised was answered in the text or the certificates; after round four the author split the work into this paper and a companion, and this is version 0.15. The author set the research direction, chose which results to pursue, relayed the outside reviews and decided each revision. The certificates (the integer Gale witness for the datum (Q(i), [-3]) and the search that shows the Schoen-only lattice condition fails for it) are exact computations in GAP 4.14.0, supplied with scripts, transcripts and rerun commands. The same version is archived on Zenodo, https://doi.org/10.5281/zenodo.23422651, with the certificates. The reviews were source-level: no human expert other than the author has checked the paper. The Q(i) statement relies on a separate, unrefereed manuscript, as stated in the paper.

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

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Version history

  1. v1Submitted by Benjamin SevernInitial depositCurrentOct 19, 2026