I have often heard the words "Moduli Stack of Elliptic Curves", but I have nowhere found a from-scratch definition of this object. I do understand the motivation: There are cusps in the moduli space that produce singularities.

For me, a stack is a generalized sheaf on a site. So I am specifically asking the following questions:

1) What is the site?

2) What are the fibers (this should be categories, even groupoids?)

I know that the upper half plane with the usual action of $\mathrm{SL}(2, \mathbb{Z})$ (which parametrizes elliptic curves) is an orbifold, which is a special stack (the site being the site of the topological space, and the fibers being more or less the automorphism group(oid)s). But is that all there is to it? Why don't peoply say "orbifold" instead of stack?

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    $\begingroup$ On your last question: why don't people say "manifold" instead of variety? $\endgroup$ – Zhen Lin Aug 8 '14 at 8:20
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    $\begingroup$ The site is something like the Zariski site. The fiber over a scheme $X$ is something like the groupoid of elliptic curves over $X$. I don't know the technical details though. $\endgroup$ – Qiaochu Yuan Aug 8 '14 at 8:34
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    $\begingroup$ You might want to take a look at Stacks Project. Section 80.3 is entitled "The moduli stack of elliptic curves". $\endgroup$ – Pece Aug 8 '14 at 8:48
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    $\begingroup$ The beautiful paper that introduces this object is Mumford's "Picard groups of moduli problems". $\endgroup$ – rfauffar Aug 8 '14 at 20:19
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    $\begingroup$ There is a very helpful expository paper by Richard Hain on the topic of Moduli spaces of elliptic curves. arxiv.org/abs/0812.1803v3 $\endgroup$ – Lorenzo Najt Jun 28 '17 at 0:12

A very good introduction is the (unfinished) book Algebraic stacks by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. You can find the finished chapters here.

Usually one considers the big étale site $\mathsf{Sch}_{ét}$ over some base scheme or base ring. But other sites are also possible, for example small fractions of the étale site in order to avoid set-theoretic problems (Tag 020M). If $S$ is some object of this site, an elliptic curve over (or, parametrized by) $S$ is a smooth proper morphism $C \to S$ with a section $\sigma : S \to C$, such that for all $s \in S$ the geometric fiber $C_{\overline{s}} \to \mathrm{Spec}(\overline{k(s)})$ is a connected curve of genus $1$ (and hence becomes an elliptic curve over $\overline{k(s)}$ with unit $\sigma_{\overline{s}}$). If $S \to T$ is a morphism and $C,D$ are elliptic curves over $S$ resp. $T$, a morphism $C \to D$ is a morphism which is compatible with the sections and which makes $$\begin{array}{cc} C & \rightarrow & D \\ \downarrow && \downarrow \\ S & \rightarrow & T \end{array}$$ cartesian. One obtains a category $M_{1,1}$ fibered over $\mathsf{Sch}_{ét}$. It is quite formal to verify that it is a stack - the moduli stack of elliptic curves. I haven't read the proof, but it seems to me that one needs the theory of elliptic curves and some of Grothendieck's theory of Hilbert schemes to verify that $M_{1,1}$ is an algebraic stack. More generally, if $g,n \geq 0$ one can define the algebraic stack $M_{g,n}$ of smooth $n$-pointed algebraic curves of genus $g$. The stack $M_g := M_{g,1}$ was introduced by Deligne and Mumford in their seminal paper "The irreducibility of the space of curves of a given genus" (pdf).

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