Abstract
AbstractLet 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.
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Algebra and Number Theory
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