R-equivalence on a diagonal cubic over Q₂(ζ₃)
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When are rational points on a cubic surface R-equivalent, meaning connected by chains of rational curves defined over the ground field? The specific diagonal cubic is X³+Y³+Z³+ζ₃T³=0 over Q₂(ζ₃), and the target is a single R-equivalence class.
Why it matters and what counts as progress
Why it matters: R-equivalence measures rational connectivity of rational points over the ground field, where diagonal cubic cases resist purely geometric arguments.
What counts as progress: Prove trivial R-equivalence for this surface or extend the argument to further diagonal cubics and ground fields.
How the problem developed
Yuri Manin Asked whether R-equivalence is trivial on the diagonal cubic surface over Q₂(ζ₃), in Cubic Forms; the question concerns this specific exceptional 2-adic surface.
- 1981 · Partial result [2]
Peter Swinnerton-Dyer Proved triviality for many good-reduction p-adic cubic surfaces, with three exceptional reduction types; Manin’s surface was not settled by that result.
- 19 Mar 2026 · Reported claim [2]
Dimitri Kanevsky, Julian Salazar, and Matt Harvey The preprint reports trivial R-equivalence for Manin’s diagonal cubic over Q₂(ζ₃), answering the named question; it also treats other all-Eckardt 2-adic cases.
AI contribution
R-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence
The diagonal cubic X³+Y³+Z³+ζ₃T³=0 over Q₂(ζ₃) has trivial R-equivalence. Gemini 3 Deep Think supplied rigorous methods for the gap-closing Lemmas 3.4–3.5 in the human-led proof.
Sources
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