Birational properties of word varieties

TL;DR

The paper studies birational properties of word varieties $S_{w,\alpha}\subset {\rm SL}(2)\times {\rm SL}(2)$ defined by $w(X,Y)=\alpha$ through the trace polynomial $P_w(s,t,u)$ and the associated trace surface $H_{w,a}$ with $a=\mathrm{tr}(\alpha)$. It shows that for non-central semisimple $\alpha$, $S_{w,\alpha}$ sits densely inside a smooth conic bundle over $H_{w,a}$, and that when $w$ lies in the commutator subgroup the setup reduces to a conic-bundle over a Markoff-type surface $M_d$ with $d=\mathrm{tr}(\alpha)-2$; this yields a precise link between rationality and the geometry of $M_d$. In the commutator case, $S_{\alpha}$ is $k$-unirational and geometrically integral, and $k$-rational exactly when $M_d$ is $k$-rational, with explicit criteria in terms of $d$ and squares in $k$. Over number fields, the Brauer–Manin obstruction governs rational points and weak approximation on smooth models birational to $S_{\alpha}$, and the methods provide a negative answer to a question of Rapinchuk–Benyash-Krivetz–Chernousov by exhibiting irrational cases; overall, the work connects word equations to conic-bundle geometry and Brauer group techniques to derive birational and arithmetic consequences.

Abstract

We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=α$, 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.

Theorems & Definitions (18)

• Proposition 1.1
• Lemma 1.2
• Example 1.3
• Remark 1.4
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Corollary 2.5
• Proposition 2.6