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

TL;DR

This paper investigates $R$-equivalence on smooth cubic surfaces over $p$-adic fields, proving triviality or exponent 2 for known cases with non-trivial universal equivalence, and confirming specific longstanding conjectures.

Contribution

It introduces new methods to analyze $R$-equivalence on cubic surfaces, establishing triviality or exponent 2 in cases previously unresolved, and confirms specific cases of longstanding questions.

Findings

$R$-equivalence is trivial or of exponent 2 on certain cubic surfaces

Confirmed triviality for specific cubic surfaces over $Q_2(Z_3)$

Provided new insights into the structure of $R$-equivalence on special cubic surfaces

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-Th\'el\`ene 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 $X^3+Y^3+Z^3+\zeta_3 T^3=0$ over $\mathbb{Q}_2(\zeta_3)$--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).