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 $h^ε$, for any positive ε.  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.