The Stacks Project


Tag 072J

95.2. 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 $$ p : \mathcal{M}_{1, 1} \longrightarrow \textit{Sch}, \quad (E, f, 0)/S \longmapsto S $$ 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.

    The code snippet corresponding to this tag is a part of the file stacks-introduction.tex and is located in lines 41–80 (see updates for more information).

    \section{Preliminary}
    \label{section-preliminary}
    
    \noindent
    Let $S$ be a scheme. An {\it 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
    \begin{enumerate}
    \item $f : E \to S$ is proper, smooth of relative dimension $1$,
    \item 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
    \item $0$ is a section of $f$.
    \end{enumerate}
    Given elliptic curves $(E, f, 0)/S$ and $(E', f', 0')/S'$ a
    {\it 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 \cite{DM} which consists in presenting
    $\mathcal{M}_{1, 1}$ as a category endowed with a functor
    $$
    p : \mathcal{M}_{1, 1} \longrightarrow \Sch, \quad
    (E, f, 0)/S \longmapsto S
    $$
    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 denis lieberman (site) on August 2, 2017 a 4:42 pm UTC

    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 Johan (site) on August 2, 2017 a 6:15 pm UTC

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

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