Arithmetic sparsity in mixed Hodge settings

TL;DR

This paper demonstrates that $S$-integral points on certain algebraic varieties with mixed Hodge structures are sparsely distributed, and applies this to count Laurent polynomials with fixed geometric properties, answering a recent open question.

Contribution

It establishes a sparsity result for $S$-integral points in mixed Hodge settings and applies it to count Laurent polynomials with fixed Newton polyhedron and determinant.

Findings

$S$-integral points are covered by subpolynomially many subvarieties.

Subpolynomial bound on the number of Laurent polynomials with fixed properties.

Answers a question by Ellenberg-Lawrence-Venkatesh.

Abstract

Let $X$ be a smooth irreducible quasi-projective algebraic variety over a number field $K$. Suppose $X$ is equipped with a $p$-adic \'{e}tale 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 $\Delta$ and fixed non-zero principal $\Delta$-determinant. Our results answer a question asked by Ellenberg-Lawrence-Venkatesh.