2022/06/22 by Kenneth Chung Tak Chiu, Chiu, Kenneth Chung Tak
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2206.11195
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 Xℂ. 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.