A nonabelian circle method

Abstract

We count integral quaternion zeros of $γ_1^2 \pm \dots \pm γ_n^2$, 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 $3n-2.$

66 pages, 0 figures. Added supplementary material by Arala, Hou, Hsu, Li, and Wang