Manin's Question on R-Equivalence for the Diagonal Cubic
Swinnerton-Dyer (1981) proved -equivalence trivial on smooth cubic surfaces over -adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic over , answering a question from Manin's Cubic Forms (1972), and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982).
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Arithmetic geometry
- Posed by
- Yuri Manin (Cubic Forms)
- Year posed
- 1972
- Years open
- 54y
- Solved
- 2026-03-19
- Model
- Gemini 3 Deep Think, AlphaEvolve
- Vendor
- Google DeepMind
- Collaborators
- Dimitri Kanevsky, Julian Salazar, Matt Harvey
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
"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.