Sparsity of Integral Points on Moduli Spaces of Varieties

TL;DR

This paper proves that integral points on certain moduli spaces of varieties are sparse, with their count growing slower than any positive power of the height bound, under specific geometric conditions.

Contribution

It establishes a new sparsity result for integral points on moduli spaces of varieties with special Hodge-theoretic properties.

Findings

Integral points grow slower than any positive power of height B.

Sparse distribution of integral points in specific polynomial families.

Results apply to varieties with zero-dimensional period mapping fibers.

Abstract

Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the number of such points of height $\leq B$ grows slower than any positive power of $B$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.