Counting rational points on smooth hypersurfaces with high degree
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912561387012096 |
|---|---|
| author | Verzobio, Matteo |
| author_facet | Verzobio, Matteo |
| contents | Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^n$ with $n\geq 4$ and degree $d\geq 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\varepsilon}(B^{n-2+\varepsilon})$. If $X$ is smooth and has degree $d\geq 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting rational points on smooth hypersurfaces with high degree Verzobio, Matteo Number Theory 11D45, 11D41 Let $X$ be a smooth projective hypersurface defined over $\mathbb{Q}$. We provide new bounds for rational points of bounded height on $X$. In particular, we show that if $X$ is a smooth projective hypersurface in $\mathbb{P}^n$ with $n\geq 4$ and degree $d\geq 50$, then the set of rational points on $X$ of height bounded by $B$ have cardinality $O_{n,d,\varepsilon}(B^{n-2+\varepsilon})$. If $X$ is smooth and has degree $d\geq 6$, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth. |
| title | Counting rational points on smooth hypersurfaces with high degree |
| topic | Number Theory 11D45, 11D41 |
| url | https://arxiv.org/abs/2503.19451 |