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

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.