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(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Algebraic geometry
- Posed by
- Jean-Louis Coliot-Thèléne
- Year posed
- 2001
- Years open
- 25y
- 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 was actually shown
The authors conjecture the period-index conjecture holds true only with corrections at small primes, and to this end proved its analog in Hodge theory.
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arXiv preprint, not yet peer-reviewed.