An example of a non-associative Moufang loop of point classes on a cubic surface

Abstract

Let $k=\mathbb{Q}_3(\theta)$, $\theta^3=1$ be a quadratic extension of 3-adic numbers. Let $V$ be a cubic surface defined over a field $k$ by the equation $T_0^3+T_1^3+T_2^3+\theta T_0^3=0$ and let $V(k)$ be a set of rational points on $V$ defined over $k$. We show that a relation on $V(k)$ modulo a prime $(1-\theta)^3$ (in a ring of integers of $k$) defines an admissible relation on a set of rational points of $V$ over $k$ and a commutative Moufang loop associated with classes of this admissible equivalence on $V(k)$ is non-associative. This answers a long standing problem that was formulated by Yu. I. Manin more than 50 years ago about existence of non-abelian quasi-groups associated with some cubic surface over some field