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

TL;DR

This paper constructs a non-associative Moufang loop of point classes on a specific cubic surface over a quadratic extension of 3-adic numbers, answering a long-standing question posed by Yu. I. Manin over 50 years ago.

Contribution

It demonstrates the existence of a non-associative Moufang loop of point classes on a cubic surface, solving a problem posed by Manin decades ago.

Findings

Established a relation on geometric points modulo (1-θ)^3

Defined an admissible relation leading to a Moufang loop

Proved the non-associativity of the constructed loop

Abstract

Let $V$ be a cubic surface defined by the equation $T_0^3+T_1^3+T_2^3+\theta T_3^3=0$ over a quadratic extension of 3-adic numbers $k=\mathbb{Q}_3(\theta)$, where $\theta^3=1$. We show that a relation on a set of geometric k-points on $V$ modulo $(1-\theta)^3$ (in a ring of integers of $k$) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes.