Use of Hilbert Schemes in Arithmetic? I'm curious about the following: What are some arithmetic application of the Hilbert Schemes?
The application of Hilbert schemes in algebraic geometry seems to be a great success, from birational geometry to enumerative geometry. However, although the Hilbert scheme is defined over $\mathbb{Z}$, I haven't seen any use of it in the arithmetic side. If you know such an example please let me know.
 A: Quite generally, whenever you ``need a moduli space'', say, polarized deformations of varieties, or spaces of morphisms, you oftentimes construct it as follows: first, you construct some family in projective space that (over-)parametrizes your data. Then, using that you've fixed some numerical invariants, you prove that the Hilbert polynomial of this projective family is constant. Then, you use the Hilbert scheme to realize this family as a subscheme of a Hilbert scheme. This is a first approximation to your moduli problem. Usually, you've overparametrized your data and need to take some appropriate stack/GIT quotient (which is usually subtle)...
Thus, in case you're interested in arithmetic moduli, say, moduli spaces of polarized Abelian varieties over $\mathbb{Z}$, moduli spaces of curves of genus $g\geq2$ over $\mathbb{Z}$, you will need the fact that the Hilbert scheme is defined over $\mathbb{Z}$.
Let me even give an application to complex geometry: when proving the existence of rational curves on (complex!) varieties, whose $K_X$ is not nef, via "bend and break", you do the following: you reduce your variety modulo positive characteristic $p$, and construct the desired rational curves on infinitely many reductions modulo $p$ using the Frobenius morphism and characteristic-$p$-methods. Then, you bound the degree of these rational curves (w.r.t. some polarization). Now, to conclude the existence of a rational curve in characteristic zero, you use the space of morphisms from $\mathbb{P}^1$ to show lifting of these curves from characteristic $p$ to characteristic zero. Here, it is essential that this space of morphisms (whose existence relies on the Hilbert scheme) is defined over some ring of integers. 
