Sorry about this question which is not on a research level. But I am very confused about this "first" example of coarse moduli space of genus 1 curves.
The moduli space I talk about here is the coarse moduli space, using the language of schemes and the moduli functor. By definition, this moduli space should be a scheme. For genus 1 curves (smooth and proper), I have the "intuition" that this moduli $\mathcal{M}_{1}$ is the "Affine line $\mathbb{A}^1$".
More precisely, as I read in the definition, the moduli functor is a function $F: Schemes \rightarrow Sets$. For a given scheme $S$, $F(S)$ is the set of all (smooth and proper) $C/S$ that has smooth (geometrical) fibers of genus 1. And the coarse moduli space is a scheme that is "closed" to the (non-exist) fine moduli space.
But, myabe it is a stupid question, what is this scheme $\mathbb{A}^1$ at all? Many notes or books don't explan this, and others even write this moduli space as $\mathbb{A}^1_k$, for a field $k$, which I don't know where this field $k$ comes from. $\mathcal{M}_{1}$ should not be an algebraic variety over a fixed field $k$, if I understand correctly.
I hope that some of you are kind to explain to me this "elementary" question. Otherwise I have a big difficulty to study the moduli space.