Sparsity of Integral Points on Moduli Spaces of Varieties

Abstract: Interesting moduli spaces don't have many integral points. More precisely, if $X$ is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of $S$-integral points on $X$ of height at most $H$ grows more slowly than $H^{\epsilon}$, for any positive $\epsilon$. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of $X$. Joint with Ellenberg and Venkatesh.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

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Columbia Automorphic Forms and Arithmetic Seminar