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The Period-Index Conjecture

Algebra · disproved

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Statement

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 kk of characteristic 00 and any d≥3d \geq 3 there is a dd-dimensional variety over kk carrying a Brauer class that violates it, for Hodge-theoretic reasons. For d=3d = 3 the construction needs no uncountability, so the conjecture fails already over Q‾\overline{\mathbf{Q}}.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Alexander Perry, using ChatGPT

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    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.

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open in Claude →open in ChatGPT →

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