A nonabelian circle method
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arXiv
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| Format: | Preprint |
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2024
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| author | Arala, Nuno Getz, Jayce R. Hou, Jiaqi Hsu, Chun-Hsien Li, Huajie Wang, Victor Y. |
| author_facet | Arala, Nuno Getz, Jayce R. Hou, Jiaqi Hsu, Chun-Hsien Li, Huajie Wang, Victor Y. |
| contents | 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 $cX^{4n-8} + O(X^{3n+\varepsilon})$ 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 $3n-2.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nonabelian circle method Arala, Nuno Getz, Jayce R. Hou, Jiaqi Hsu, Chun-Hsien Li, Huajie Wang, Victor Y. Number Theory 11D85, 11F70, 11P05, 11P55 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 $cX^{4n-8} + O(X^{3n+\varepsilon})$ 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 $3n-2.$ |
| title | A nonabelian circle method |
| topic | Number Theory 11D85, 11F70, 11P05, 11P55 |
| url | https://arxiv.org/abs/2407.11804 |