Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Banach Center Publications / All volumes

Abstract

Let $V$ be the cubic surface defined by the equation $T_0^3+T_1^3+T_2^3+\theta T_3^3=0$ over the quadratic extension $k=\mathbb {Q}_3(\theta )$ of the 3-adic rationals, where $\theta ^3=1$. We show that a relation on $V(k)$ modulo $(1-\theta )^3$ (in the ring of integers of $k$) defines a universal (i.e. the finest possible admissible) equivalence relation on the set of rational points of $V$. This is the continuation of the author’s recent work that provided the first example where the commutative Moufang loop of point classes on a cubic surface is not associative.