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

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

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algebraMar 19, 2026Significance 15/100Registry: unreviewed

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

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

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

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