Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910931527663616 |
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| author | Alfaraj, Abdulmuhsin |
| author_facet | Alfaraj, Abdulmuhsin |
| contents | We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of $\mathbb{P}^{p-1}$, where $p$ is the characteristic of $F$, viewed as a compactification of appropriate $F$-wound groups to illustrate new phenomena appearing in the function field setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$ Alfaraj, Abdulmuhsin Number Theory Algebraic Geometry 14G05 (Primary) 14G12, 14G17 (Secondary) We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of $\mathbb{P}^{p-1}$, where $p$ is the characteristic of $F$, viewed as a compactification of appropriate $F$-wound groups to illustrate new phenomena appearing in the function field setting. |
| title | Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$ |
| topic | Number Theory Algebraic Geometry 14G05 (Primary) 14G12, 14G17 (Secondary) |
| url | https://arxiv.org/abs/2505.04562 |