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

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 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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