The Period-Index Conjecture
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field of characteristic and any there is a -dimensional variety over carrying a Brauer class that violates it, for Hodge-theoretic reasons. For the construction needs no uncountability, so the conjecture fails already over .
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Algebraic geometry
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-04
- Model
- ChatGPT
- Vendor
- OpenAI
- Collaborators
- Alexander Perry
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper's AI disclosure is unusually specific about a partial success. Prompted to find a counterexample by the Hodge-theoretic strategy, ChatGPT produced an example that was flawed, but whose shape survived into the final solution: a quotient of the same form, with an abelian surface, a genus two curve and the group Z/4. The author refined that into the working construction.
Verification
arXiv preprint, not yet peer-reviewed.