Arithmetic sparsity in mixed Hodge settings
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913989141725184 |
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| author | Chiu, Kenneth Chung Tak |
| author_facet | Chiu, Kenneth Chung Tak |
| contents | Let $X$ be a smooth irreducible quasi-projective algebraic variety over a number field $K$. Suppose $X$ is equipped with a $p$-adic étale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on the complex analytification of $X_{\mathbb{C}}$. We prove that the $S$-integral points in $X$ are covered by subpolynomially many geometrically irreducible $K$-subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe-Maculan and Ellenberg-Lawrence-Venkatesh. As an application, we prove that there are subpolynomially many $S$-integral Laurent polynomials with fixed reflexive Newton polyhedron $Δ$ and fixed non-zero principal $Δ$-determinant. Our results answer a question asked by Ellenberg-Lawrence-Venkatesh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_11195 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arithmetic sparsity in mixed Hodge settings Chiu, Kenneth Chung Tak Number Theory Algebraic Geometry Let $X$ be a smooth irreducible quasi-projective algebraic variety over a number field $K$. Suppose $X$ is equipped with a $p$-adic étale local system compatible with an admissible graded-polarized variation of mixed Hodge structures on the complex analytification of $X_{\mathbb{C}}$. We prove that the $S$-integral points in $X$ are covered by subpolynomially many geometrically irreducible $K$-subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe-Maculan and Ellenberg-Lawrence-Venkatesh. As an application, we prove that there are subpolynomially many $S$-integral Laurent polynomials with fixed reflexive Newton polyhedron $Δ$ and fixed non-zero principal $Δ$-determinant. Our results answer a question asked by Ellenberg-Lawrence-Venkatesh. |
| title | Arithmetic sparsity in mixed Hodge settings |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2206.11195 |