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

Abstract

Abstract Let V be a cubic surface defined by the equation $$T_0^3+T_1^3+T_2^3+\theta T_3^3=0$$ T 0 3 + T 1 3 + T 2 3 + θ T 3 3 = 0 over a quadratic extension of 3-adic numbers $$k=\mathbb {Q}_3(\theta )$$ k = Q 3 ( θ ) , where $$\theta ^3=1$$ θ 3 = 1 . We show that a relation on a set of geometric k-points on V modulo $$(1-\theta )^3$$ ( 1 - θ ) 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.