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arXiv · 2504.15461

Birational properties of word varieties

Tatiana Bandman·Boris Kunyavskii·Alexei N. Skorobogatov
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Abstract

We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=\alpha$, where $w$ is a word in two letters, is closely related to an explicit smooth conic bundle over the associated `trace surface' in the 3-dimensional affine space. When $w$ is the commutator word, we show that this variety can be irrational if the ground field $k$ is not algebraically closed, answering a question of Rapinchuk, Benyash-Krivetz, and Chernousov. When $k$ is a number field, it satisfies weak approximation with the Brauer--Manin obstruction.

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Tatiana Bandman, Boris Kunyavskii, Alexei N. Skorobogatov. 2025-04-21. Birational properties of word varieties. https://arxiv.org/abs/2504.15461

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