TITLE: What does it mean for points of the modular curve $X(N)$ to be "defined over $\mathbb{F}_p$"?
QUESTION [7 upvotes]: I'm trying to study a collection of elliptic curves over some fixed finite field $\mathbb{F}_p$. By browsing the literature and discussing with my supervisor, it seems like it will be fruitful to study some sort of a modular curve.
I am familiar with the classic modular curves $X(N),X_0(N),X_1(N)$, etc. through the first three chapters of Diamond and Shurman's A First Course In Modular Forms. I understand that these curves are Riemann surfaces and form a moduli space for isomorphism classes of elliptic curves over $\mathbb{C}$.
In this paper that I am reading, the authors speak of points of $X_1(N)$ that are defined over $\mathbb{F}_p$ and they use the notation $X_1(N)(\mathbb{F}_p)$ which makes it seem like they're probably using the language of schemes.
I am aware of the books (or long papers):

Les schemas de modules de courbes elliptiques by Deligne-Rapoport (1973)
Arithmetic Moduli of Elliptic Curves by Katz-Mazur (1985)

I know the basic definitions of schemes from Hartshorne's Algebraic Geometry but I do not know many results beyond their first properties. Both of these books address the topic of reduction mod $p$ at some point and I believe this is exactly what I'm looking for. However, as I'm just beginning my study on this topic, I'm finding it hard to digest the discussions, or to even locate the relevant discussions, in either book, especially the first one, which is written in French, and seems to be the canonical reference on this subject by my search online.
Any of the following responses will be helpful to me:

A rough sketch motivating what it means to have points of $X(N),X_0(N),X_1(N)$ defined over $\mathbb{F}_p$.
Reference to a specific page or section in Deligne-Rapoport or Katz-Mazur addressing my first point.
Reference to a source outside of Deligne-Rapoport and Katz-Mazur addressing my first point. Ideally, the source should be in English.

REPLY [3 votes]: There is an excellent answer above, so I'm not going to address the general situation - instead I'll provide a different (much more explicit) description when thinking about $X_1(N)$ and discuss the case when $N = 5$ which (in my opinion) gives a bit more intuition that grappling with the general setup in the (pretty difficult) material of KM and DR.
Recall that the moduli problem is set up as follows (at least in moral terms) for $Y_1(N)$:

Does there exist a curve, $X_1(N)$, over $\mathbb{Q}$ (can we make sure it's defined over $\mathbb{Q}$ with integer coefficients and make sure bad stuff doesn't happen when reducing mod $p$ for $p$ not dividing $N$?) such that for each $K/\mathbb{Q}$, each point $Q \in X_1(N)(K)$ corresponds to a pair $(E, P)$ where $E/K$ is an elliptic curve and $P$ is a $K$-rational point on $E$ of exact order $N$?

The answer to the above question is "yes when $N > 3$" (and note that when you take $K = \mathbb{C}$ this is exactly what you know!). With $X_0$ and $X$ you need to replace $K$-rational with "galois stable cyclic subgroup (or isogeny over $K$)" and "$K$-rational basis $\{P_1,P_2\}$ and some stuff with the Weil pairing" respectively.
Now for something explicit. Lets compute! $X_1(5)$ (well, of course this is just $\mathbb{P}^1$, but the moduli interpretation will also come).
I'll follow Silverman's The Arithmetic of Elliptic Curves exercise $8.13$. Take a pair $(E, P)$ where $E/K$ is an elliptic curve ($K$ has characteristic $0$) and $P \in E(K)$ has order $5$. We can assume that $E$ is given by a Weierstrass equation
$$E : y^2 + u xy + vy = x^3 + vx^2$$
and $P = (0,0)$ (prove this for general $N \geq 4$ - what about the characteristic). Then for $5P = O$ we must have $3P = -2P$. In particular we must have
$$(-u + 1 : u - v - 1 : 1) = (-v : 0 : 1) $$
i.e., $P$ has order $5$ if and only if $v = u-1$. So we have $X_1(5) \cong \mathbb{P}^1$ and the moduli interpretation takes
$$ [u:1] \mapsto (E_u, (0,0) )$$
where $E_u : y^2 + uxy + (u-1)y = x^3 + (u-1)x^2$.
So now: (1) can you mimic this to work over $\mathbb{F}_p$ where $p \neq 5$?, (2) can you find the hole when $p = 5$?
A bit harder is to use this exact construction to show that when $\operatorname{K}$ has characteristic not diving $N$ there is a curve $X_1(N)$ which works. A bit harder than that is to show that the "same" curve works for each characteristic (this is more or less "the moduli problem is representable over $\mathbb{Z}[1/N]$).
The moral is just that this is not so scary as long as $K$ does not have characteristic $p$!