论文标题
混合霍奇设置中的算术稀疏性
Arithmetic sparsity in mixed Hodge settings
论文作者
论文摘要
让$ x $成为一个平稳的不可约准标记代数$ k $。假设$ x $配备了$ p $ adiCétale本地系统兼容，与$ x _ {\ mathbb {c}} $的复杂分析化的混合Hodge结构的可允许分级偏振变化。我们证明，$ x $中的$ s $综合点在许多几何不可约$ k $ - subvarieties上涵盖了许多几何不可约的$ k $ - subvarieties，每个都位于由混合Hodge结构的变化所产生的混合期映射的纤维中。这是基于Brunebarbe-Maculan和Ellenberg-Lawrence-Venkatesh的最新作品。作为一个应用程序，我们证明，在多个$ s $ intemental laurent laurent多项式上有固定反射性牛顿多面体$Δ$和固定的非零主$δ$δ$ - 确定性。我们的结果回答了Ellenberg-Lawrence-Venkatesh问的问题。
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.
