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Manin's Question on R-Equivalence for the Diagonal Cubic

Algebra · posed by Yuri Manin (Cubic Forms), 1972 · solved

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Statement

Swinnerton-Dyer (1981) proved RR-equivalence trivial on smooth cubic surfaces over pp-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic over Q3\mathbb{Q}_3, answering a question from Manin's Cubic Forms (1972), and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982).

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

  1. proof attempt · #1

    Dimitri Kanevsky, Julian Salazar and Matt Harvey, using Gemini 3 Deep Think, AlphaEvolve

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

    "This is the first in a series of works derived from a year of interactions with generative AI models such as AlphaEvolve and Gemini 3 Deep Think, with the latter proving many of our lemmas." The paper devotes a section to the timeline and nature of the AI use; the authors are career AI researchers at Google DeepMind, and Kanevsky posed one of the resolved cases himself in 1982.

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