A nonabelian circle method

We count integral quaternion zeros of ${\gamma }_1^2\pm \cdots \pm {\gamma }_n^2$, giving an asymptotic when $n\ge 9$, and a likely near-optimal bound when $n=8$. To do so, we introduce a new, nonabelian delta symbol method, which is of independent interest. Our asymptotic at height $X$ takes the form $c{X}^{4n-8}+O(X^{3n+\epsilon })$ for suitable $c\in \mathbb{C}$ and any $\epsilon >0.$ We construct special subvarieties implying that, in general, $3n+\epsilon$ can be at best improved to $3n-2.$