$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence
This resolves specific cases in algebraic geometry concerning rational points on cubic surfaces, though it appears incremental as it builds on prior work by Swinnerton-Dyer and others.
The paper tackles the problem of determining when R-equivalence is trivial on smooth cubic surfaces over p-adic fields, focusing on cases with non-trivial universal equivalence that were previously intractable. The authors prove that for 2-adic surfaces with all-Eckardt reductions, R-equivalence is trivial or of exponent 2, and confirm triviality for explicit cases including the diagonal cubic over Q₂(ζ₃), answering a long-standing question.
Let $V$ be a smooth cubic surface over a $p$-adic field $k$ with good reduction. Swinnerton-Dyer (1981) proved that $R$-equivalence is trivial on $V(k)$ except perhaps if $V$ is one of three special types--those whose $R$-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces $V$ currently known to have non-trivial universal equivalence. Beyond being intractable to Swinnerton-Dyer's approach, we observe that if these surfaces also had non-trivial $R$-equivalence, they would contradict Colliot-Thélène and Sansuc's conjecture regarding the $k$-rationality of universal torsors for geometrically rational surfaces. By devising new methods to study $R$-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions (the third special type, which contains every existing case of non-trivial universal equivalence), $R$-equivalence is trivial or of exponent 2. For the explicit cases, we confirm triviality: the diagonal cubic $X^3+Y^3+Z^3+ζ_3 T^3=0$ over $\mathbb{Q}_2(ζ_3)$--answering a long-standing question of Manin's (Cubic Forms, 1972)--and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982). 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. We disclose the timeline and nature of their use towards this paper, and describe our broader AI-assisted research program in a companion report (in preparation).