Manin's Question on R-Equivalence for the Diagonal Cubic
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Statement
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).
Context
A question standing since Manin's Cubic Forms (1972) inside Swinnerton-Dyer's classification, an established target of the arithmetic of cubic surfaces with a five-decade paper trail.
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"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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