The Stacks project

104.2 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

\[ \xymatrix{ E \ar[rr]_\alpha \ar[d]^ f & & E' \ar[d]_{f'} \\ S \ar@/^5ex/[u]^0 \ar[rr]^ a & & S' \ar@/_5ex/[u]_{0'} } \]

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

\[ p : \mathcal{M}_{1, 1} \longrightarrow \mathit{Sch}, \quad (E, f, 0)/S \longmapsto S \]

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.


Comments (2)

Comment #2754 by on

E in the triple appearing in the first line is not a scheme but a Deligne- Rapoport (DR) semistable genus-1 curve. This statement appears in Brian Conrad's kmpaper. He gives his source as: Les schemes de modules des courbes elliptiques in Modular Functions of One Variable II, Springer Lecture Notes in Mathematics 349 (1973) pp.143-316.

Comment #2755 by on

Dear Denis Lieberman, actually here we only do elliptic curves and hence we get schemes.


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