The Circle Method for Quadrics over Function Fields
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912998888570880 |
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| author | Mettasch, Johanna |
| author_facet | Mettasch, Johanna |
| contents | We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$. Viewing these points as morphisms $\mathbb{P}^1 \to X$ of fixed degree, we obtain exact formulas for their number depending on the parity of $n$ and on the determinant of the quadratic form defining $X$, including secondary terms in some cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02067 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Circle Method for Quadrics over Function Fields Mettasch, Johanna Number Theory 11D45 (Primary) 11E12, 11P55, 11T24, 14G05 (Secondary) We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$. Viewing these points as morphisms $\mathbb{P}^1 \to X$ of fixed degree, we obtain exact formulas for their number depending on the parity of $n$ and on the determinant of the quadratic form defining $X$, including secondary terms in some cases. |
| title | The Circle Method for Quadrics over Function Fields |
| topic | Number Theory 11D45 (Primary) 11E12, 11P55, 11T24, 14G05 (Secondary) |
| url | https://arxiv.org/abs/2604.02067 |