TITLE: Moduli of curves over finite field
QUESTION [5 upvotes]: This is related to this question. I learnt about moduli problem mainly with the book Harris and Morrison. Therefore, I have only seen the construction of moduli spaces $M_{g}$ over $\mathbb{C}$. But now I want to change the base field. It does not really make sense to me to take $\mathbb{F}_q$-points as described in the comments of that post, because $M_g$ is a $\text{Spec}(\mathbb{C})$-scheme not a $\text{Spec}(\mathbb{F}_q)$-scheme. So how should I understand $M_{g}$ as a coarse moduli space properly?

REPLY [10 votes]: As requested, I am posting my comment as an answer.  Mumford discusses this in the preface to the first edition of "Geometric Invariant Theory".
Already in "Geometric Invariant Theory" (which was Mumford's thesis), Mumford constructed the coarse moduli scheme $M_g$ over $\text{Spec}\ \mathbb{Z}.$ You can read about what this means in Mumford's book. The great irony of GIT is that, although GIT was largely motivated by this problem, in fact Mumford constructs $M_g$ over $\text{Spec}\ \mathbb{Z}$ without using GIT (precisely because neither had Haboush yet extended GIT to char $p$, nor had Seshadri extended GIT to mixed characteristic).