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arXiv · 2407.11804

A nonabelian circle method

Nuno Arala·Jayce R. Getz·Jiaqi Hou·Chun-Hsien Hsu·Huajie Li·Victor Y. Wang
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Abstract

We count integral quaternion zeros of $\gamma_1^2 \pm \dots \pm \gamma_n^2$, 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 $cX^{4n-8} + O(X^{3n+\varepsilon})$ for suitable $c \in \mathbb{C}$ and any $\varepsilon>0.$ We construct special subvarieties implying that, in general, $3n+\varepsilon$ can be at best improved to $3n-2.$

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Nuno Arala, Jayce R. Getz, Jiaqi Hou, Chun-Hsien Hsu, Huajie Li, Victor Y. Wang. 2024-07-16. A nonabelian circle method. https://arxiv.org/abs/2407.11804

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