Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912121388793856 |
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| author | Binyamini, Gal Cluckers, Raf Kato, Fumiharu |
| author_facet | Binyamini, Gal Cluckers, Raf Kato, Fumiharu |
| contents | Let $C\subset{\mathbb P}_K^2$ be an algebraic curve over a number field $K$, and denote by $d_K$ the degree of $K$ over ${\mathbb Q}$. We prove that the number of $K$-rational points of height at most $H$ in $C$ is bounded by $c d^{2}H^{2d_K/d}(\log H)^κ$ where $c,κ$ are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties.
The quadratic dependence on $d$ in the bound as well as the exponent of $H$ are optimal; the novel aspect is the quadratic dependence on $d$ which answers a question raised by Salberger. We derive new results on Heath-Brown's and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case $K={\mathbb Q}$. The proofs however are of a completely different nature, replacing the real analytic approach previously used by the $p$-adic determinant method. The optimal dependence on $d$ is achieved using a technical improvement in the treatment of high multiplicity points on mod $p$ reductions of algebraic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields Binyamini, Gal Cluckers, Raf Kato, Fumiharu Number Theory Algebraic Geometry Let $C\subset{\mathbb P}_K^2$ be an algebraic curve over a number field $K$, and denote by $d_K$ the degree of $K$ over ${\mathbb Q}$. We prove that the number of $K$-rational points of height at most $H$ in $C$ is bounded by $c d^{2}H^{2d_K/d}(\log H)^κ$ where $c,κ$ are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties. The quadratic dependence on $d$ in the bound as well as the exponent of $H$ are optimal; the novel aspect is the quadratic dependence on $d$ which answers a question raised by Salberger. We derive new results on Heath-Brown's and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case $K={\mathbb Q}$. The proofs however are of a completely different nature, replacing the real analytic approach previously used by the $p$-adic determinant method. The optimal dependence on $d$ is achieved using a technical improvement in the treatment of high multiplicity points on mod $p$ reductions of algebraic curves. |
| title | Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2401.03982 |