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, E
Chernousov. When $k$ is a number field, it satisfies weak approximation with
the Brauer–Manin obstruction.

Questo articolo esplora i giri e le loro implicazioni.

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