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arXiv · 2104.05118

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

Dimitri Kanevsky
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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.

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Dimitri Kanevsky. 2021-04-11. An example of a non-associative Moufang loop of point classes on a cubic surface. https://arxiv.org/abs/2104.05118

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