$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

• URL: http://arxiv.org/abs/2603.19215v1
• Date: Thu, 19 Mar 2026 17:57:38 GMT
• Title: $R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence
• Abstract summary: We consider all surfaces $V$ currently known to have non-trivial $R$-equivalence.<n>By devising new methods to study $R$-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions, $R$-equivalence is trivial or of exponent 2.
• Score: 2.809000787228893
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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).

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