The Stacks project

105.2 Preliminary

Let $S$ be a scheme. An elliptic curve over $S$ is a triple $(E, f, 0)$ where $E$ is a scheme and $f : E \to S$ and $0 : S \to E$ are morphisms of schemes such that

1. $f : E \to S$ is proper, smooth of relative dimension $1$,

2. for every $s \in S$ the fibre $E_ s$ is a connected curve of genus $1$, i.e., $H^0(E_ s, \mathcal{O})$ and $H^1(E_ s, \mathcal{O})$ both are $1$-dimensional $\kappa (s)$-vector spaces, and

3. $0$ is a section of $f$.

Given elliptic curves $(E, f, 0)/S$ and $(E', f', 0')/S'$ a morphism of elliptic curves over $a : S \to S'$ is a morphism $\alpha : E \to E'$ such that the diagram

is commutative and the inner square is cartesian, in other words the morphism $\alpha $ induces an isomorphism $E \to S \times _{S'} E'$. We are going to define the stack of elliptic curves $\mathcal{M}_{1, 1}$. In the rest of the Stacks project we work out the method introduced in Deligne and Mumford's paper [DM] which consists in presenting $\mathcal{M}_{1, 1}$ as a category endowed with a functor

This means you work with fibred categories over the categories of schemes, topologies, stacks fibred in groupoids, coverings, etc, etc. In this chapter we throw all of that out of the window and we think about it a bit differently – probably closer to how the initiators of the theory started thinking about it themselves.