Counting rational points on smooth quartic and quintic surfaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909984767344640 |
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| author | Andreaus, Lorenzo |
| author_facet | Andreaus, Lorenzo |
| contents | Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in $X$. Assuming a suitable hypothesis on the size of the rank of Abelian varieties, we show that $N_{X^{\prime}}(B)\ll_{K,d,\varepsilon} B^{4/3+\varepsilon}$ for any fixed $\varepsilon>0$. This improves an unconditional bound from Salberger for $d=4$ and $d=5$. The proof, based on an argument of Heath-Brown, consists of cutting $X$ by projective planes and using a uniform version of Faltings's Theorem, due to Dimitrov, Gao, and Habegger, to bound the number of rational points on the plane sections of $X$. More generally, we prove that if $X\subseteq \mathbb{P}^n$ is a non-degenerate non-uniruled smooth projective surface defined over $K$, then $N_{X^{\prime}}(B)\ll_{K,n,d,\varepsilon}B^{\frac{n+1}{n}+\varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting rational points on smooth quartic and quintic surfaces Andreaus, Lorenzo Number Theory Algebraic Geometry 11G35, 11G50, 14G05 Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in $X$. Assuming a suitable hypothesis on the size of the rank of Abelian varieties, we show that $N_{X^{\prime}}(B)\ll_{K,d,\varepsilon} B^{4/3+\varepsilon}$ for any fixed $\varepsilon>0$. This improves an unconditional bound from Salberger for $d=4$ and $d=5$. The proof, based on an argument of Heath-Brown, consists of cutting $X$ by projective planes and using a uniform version of Faltings's Theorem, due to Dimitrov, Gao, and Habegger, to bound the number of rational points on the plane sections of $X$. More generally, we prove that if $X\subseteq \mathbb{P}^n$ is a non-degenerate non-uniruled smooth projective surface defined over $K$, then $N_{X^{\prime}}(B)\ll_{K,n,d,\varepsilon}B^{\frac{n+1}{n}+\varepsilon}$. |
| title | Counting rational points on smooth quartic and quintic surfaces |
| topic | Number Theory Algebraic Geometry 11G35, 11G50, 14G05 |
| url | https://arxiv.org/abs/2511.07060 |