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\begin{document}

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\section*{moduli stack of elliptic curves}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{DescriptionOverTheComplexNumbers}{Description over the complex numbers}\dotfill \pageref*{DescriptionOverTheComplexNumbers} \linebreak
\noindent\hyperlink{upper_half_plane}{Upper half plane}\dotfill \pageref*{upper_half_plane} \linebreak
\noindent\hyperlink{the_naive_moduli_space_and_its_problems}{The naive moduli space and its problems}\dotfill \pageref*{the_naive_moduli_space_and_its_problems} \linebreak
\noindent\hyperlink{moduli_space_of_framed_elliptic_curves}{Moduli space of framed elliptic curves}\dotfill \pageref*{moduli_space_of_framed_elliptic_curves} \linebreak
\noindent\hyperlink{moduli_stackorbifold_of_elliptic_curves}{Moduli stack/orbifold of elliptic curves}\dotfill \pageref*{moduli_stackorbifold_of_elliptic_curves} \linebreak
\noindent\hyperlink{compactified_moduli_stack}{Compactified moduli stack}\dotfill \pageref*{compactified_moduli_stack} \linebreak
\noindent\hyperlink{DescriptionOverGeneralSchemes}{Description over general schemes}\dotfill \pageref*{DescriptionOverGeneralSchemes} \linebreak
\noindent\hyperlink{InEInfinityGeometry}{As a derived scheme in $E_\infty$-geometry}\dotfill \pageref*{InEInfinityGeometry} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{finite_covers}{Finite covers}\dotfill \pageref*{finite_covers} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{EulerCharacteristic}{Euler characteristic}\dotfill \pageref*{EulerCharacteristic} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{moduli stack of elliptic curves} is a [[moduli stack]] of [[elliptic curves]], hence a [[stack]] $\mathcal{M}_{ell}$ such that for $X$ any other suitable [[space]], the [[groupoid]] of maps $X \to \mathcal{M}_{ell}$ and [[homotopies]] between them is [[equivalence of groupoids|equivalent]] to that of $X$-parameterized [[elliptic curves]] with equivalences between these. (There are some variants of $\mathcal{M}_{ell}$ corresponding to the choice of which singularities and degeneracies of elliptic curves are taken into account.)

This is formalized in [[algebraic geometry]], hence $X$ here is a [[scheme]] over the [[integers]] in general. The moduli stack always has a tautological construction as a ``[[sheaf]] of [[groupoids]]'' (a [[stack]], whence the name) over the [[site]] of [[affine schemes]], given by sending any $Spec(R)$ to the groupoid of suitable [[elliptic curves]] over $Spec(R)$. For concrete computations it typically helps to know that the moduli stack of elliptic curbes is [[representable functor|represented]] by a [[geometric stack]], [[Isbell duality|dually]] given by a [[Hopf algebroid]].

The moduli stack $\mathcal{M}_{ell}$ has a [[Deligne-Mumford compactification|compactification]] $\mathcal{M}_{\overline{ell}}$ obtained by adding the [[nodal cubic curve]], and often (but not always) this compactified version is the default meaning of ``moduli stack of elliptic curves''. Adding also the [[cuspidal cubic curve]] and hence all [[cubic curves]] produces the full moduli stack $\mathcal{M}_{cub}$ of cubic curves, inside which $\mathcal{M}_{ell}$ sits as the locus of non-singular curves.

Since an [[elliptic curve]] is a [[arithmetic genus|genus-1]] [[algebraic curve]] with a marked point (the neutral element of the [[group]] structure), $\mathcal{M}_{ell}$ is equivalently the [[moduli stack of algebraic curves]] for genus $g = 1$ with $n = 1$ punctures, and as such is often equivalently written

\begin{displaymath}
\mathcal{M}_{1,1} = \mathcal{M}_{ell}
  \,.
\end{displaymath}
A special class of cases which is much simpler than the general case but still of paramount interest is the moduli stack of elliptic curves over the [[complex numbers]], hence of maps $Spec(\mathbb{C}) \to \mathcal{M}_{ell}$. These are just [[complex manifold|complex]] [[tori]]/[[Riemannian manifolds]] of [[genus of a surface|genus]] 1 which may be identified with quotients of the [[complex plane]] by a framed [[lattice]] well-defined up to [[Möbius transformations]], and so in this case the moduli stack of elliptic curves is just the [[homotopy quotient]] (the [[orbifold]] quotient) of the [[upper half plane]] by the [[action]] of the [[modular group]]. (This is equivalently the [[moduli space of curves]] $\mathcal{M}_{(1,1)}$ which in turn is a quotient of the [[Teichmüller space]] $\mathcal{T}_{(1,1)}$.) This case is considered below in

\begin{itemize}%
\item \emph{\hyperlink{DescriptionOverTheComplexNumbers}{Description over the complex numbers}}.

\end{itemize}
Below that is the

\begin{itemize}%
\item \emph{\hyperlink{DescriptionOverGeneralSchemes}{Description over general schemes}}.

\end{itemize}
as an [[algebraic stack]] $\mathcal{M}_{ell}$. This is still not the most refined description: by the [[Goerss-Hopkins-Miller theorem]] the assignment to an [[elliptic curve]] of its [[elliptic spectrum]] lifts the ordinary [[structure sheaf]] $\mathcal{O}$ of $\mathcal{M}_{ell}$ to a higher structure sheaf $\mathcal{O}^{top}$ of [[E-∞ rings]] in a way that makes $(\mathcal{M}_{ell}, \mathcal{O}^{op})$ a [[spectral Deligne-Mumford stack]]. The [[global sections]] of this structure sheaf yield the [[spectrum]] [[tmf]] of [[topological modular forms]]:

\begin{itemize}%
\item \emph{\hyperlink{InEInfinityGeometry}{As a derived scheme in E-infinity geometry}}

\end{itemize}
See also \emph{[[A Survey of Elliptic Cohomology - elliptic curves]]} for more.

\hypertarget{DescriptionOverTheComplexNumbers}{}\subsection*{{Description over the complex numbers}}\label{DescriptionOverTheComplexNumbers}

An [[elliptic curve]] $E \to Spec(\mathbb{C})$ over the [[complex numbers]] is determined, up to non-canonical [[isomorphism]], by its [[j-invariant]]

\begin{displaymath}
j(E) \in \mathbb{C}
  \,.
\end{displaymath}
Here every [[complex number]] appears as a value, and therefore the moduli space of elliptic cuves a priori is not [[compact topological space|compact]].

A [[compactification]] of the moduli space is obtained by including also elliptic curves with [[nodal singularity]].

\hypertarget{upper_half_plane}{}\subsubsection*{{Upper half plane}}\label{upper_half_plane}

The [[upper half plane]] $\mathfrak{h}$ is in [[bijection]] with framed [[lattices]] in the [[complex plane]] $\mathbb{C}$, which in turn is in bijection with [[isomorphism]] classes of framed elliptic curves over $\mathbb{C}$

\begin{displaymath}
\mathfrak{h}
  \simeq
  \{framed\;lattices\;in\;\mathbb{C}\}
  \simeq
  \{framed\;elliptic\;curves\;over\;\mathbb{C}\}/_\sim
\end{displaymath}
and we have

\begin{displaymath}
\{elliptic\;curves\;over\;\mathbb{C}\}_\sim
  \simeq
  \mathfrak{h}/{SL_2(\mathbb{Z})}
\end{displaymath}
where the [[special linear group]] over the [[integers]]

\begin{displaymath}
SL_2(\mathbb{Z}) = \left\{ \left(\itexarray{a & b \\ c & d }\right)| a d - c d = 1\right\}
\end{displaymath}
[[action|acts]] as the [[modular group]] by [[Möbius transformations]]

\begin{displaymath}
\tau \mapsto \frac{a \tau + b}{c \tau + d}
  \,.
\end{displaymath}
\hypertarget{the_naive_moduli_space_and_its_problems}{}\subsubsection*{{The naive moduli space and its problems}}\label{the_naive_moduli_space_and_its_problems}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})
\end{displaymath}
for the plain [[quotient]] of the [[upper half plane]] by the above group action.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{homolorphic family of [[elliptic curves]]} over a [[complex manifold]] $T$ is

\begin{itemize}%
\item a [[holomorphic function]] $\pi : X \to T$


\item together with a [[section]] $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an [[elliptic curve]] (using the first definition above).



\end{itemize}
\end{defn}
For every family

\begin{displaymath}
\itexarray{
    X \\ \downarrow^{\mathrlap{\pi}} \\ T 
  }
\end{displaymath}
we would like to have $F \to M_{1,1}$ such that there is a [[pullback]]

\begin{displaymath}
\itexarray{
    X \simeq \phi^* F &\longrightarrow& F
    \\
    \downarrow & & \downarrow
    \\
    T &\stackrel{\phi}{\longrightarrow}& M_{1,1}
  }
\end{displaymath}
where

\begin{displaymath}
\phi:   t \mapsto [X_t, s(t)]
\end{displaymath}
such that

\begin{itemize}%
\item $\phi : T \to M_{1,1}$ is a [[holomorphic map]]


\item every [[holomorphic map]] $T \to M_{1,1}$ corresponds to a family over $t$;


\item there is a universal family over $M_{1,1}$



\end{itemize}
This is \emph{impossible} . One can construct explicit counterexamples. These counterexamples involve [[elliptic curves]] with nontrivial [[automorphism]]s.

For instance

\begin{displaymath}
\{
    (x,y,z) \in \mathbb{P}^2 \times X : 
    y^2 = x(x-1)(x-\lambda)
  \}
  \to
  X := \mathbb{P}^1 - \{0,1,\infty\}
\end{displaymath}
\begin{quote}%
but see the discussion at [[moduli space]] for a discussion of the statement ``it's the automorphisms that prevent the [[moduli space]] from existing''


\end{quote}
\hypertarget{moduli_space_of_framed_elliptic_curves}{}\subsubsection*{{Moduli space of framed elliptic curves}}\label{moduli_space_of_framed_elliptic_curves}

consider

\begin{displaymath}
\mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}
\end{displaymath}
given by

\begin{displaymath}
(m,n) : 
  (z,\tau)
  \mapsto
  (z + m \tau + n, \tau)
\end{displaymath}
Then consider the family

\begin{displaymath}
\itexarray{
     E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h}
    \\ \downarrow \\ \mathfrak{h}
  }
\end{displaymath}
is a family of [[elliptic curve]]s over $\mathfrak{h}$

and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with

\begin{displaymath}
\Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau
\end{displaymath}
is a family of framed elliptic curves.

\begin{prop}
\label{}\hypertarget{}{}
The space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a [[fine moduli space]] for [[framed elliptic curves]].

\end{prop}
Consider any map $\phi : T \to \mathfrak{h}$

with pullback of the universal family

\begin{displaymath}
\itexarray{
     X \stackrel{?}{\to} \phi^* E &\to & E
    \\
    \downarrow && \downarrow
    \\
    T &\stackrel{\phi}{\to}&
    \mathfrak{h} 
  }
\end{displaymath}
\textbf{claim} for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose [[differential form|1-forms]] $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.

Notice that \emph{locally} every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$). So

\begin{displaymath}
\itexarray{
    && \mathfrak{h}
    \\
    && \downarrow^{SL_2(\mathbb{Z})}
    \\
    M_{1,1} &\stackrel{Id}{\to}&
    M_{1,1}
  }
\end{displaymath}
at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$

isn't locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.

\hypertarget{moduli_stackorbifold_of_elliptic_curves}{}\subsubsection*{{Moduli stack/orbifold of elliptic curves}}\label{moduli_stackorbifold_of_elliptic_curves}

\begin{defn}
\label{}\hypertarget{}{}
Consider the global [[quotient stack]] [[orbifold]]

\begin{displaymath}
\mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
\end{displaymath}
of the upper half plane by the [[action]] of the [[special linear group]] over the [[integers]].

\end{defn}
This is the [[moduli stack]] of elliptic curves.

\hypertarget{compactified_moduli_stack}{}\subsubsection*{{Compactified moduli stack}}\label{compactified_moduli_stack}

Consider the complex analytic parameterization over the [[annulus]]

\begin{displaymath}
\{q \in \mathbb{C} | 0 \lt {\vert q \vert} \lt 1 \}
\end{displaymath}
of elliptic curves

\begin{displaymath}
E_q \coloneqq \mathbb{C}/q^{\mathbb{Z}}
  \,.
\end{displaymath}
This has an extension to the origin, where $E_0$ is a [[nodal curve]]. Algebraically, in a [[formal neighbourhood]] of the origin, hence over $Spec(\mathbb{Z}[ [q] ])$, this is the [[Tate curve]].

e.g. (\hyperlink{Lurie}{Lurie, section 4.3}).

\hypertarget{DescriptionOverGeneralSchemes}{}\subsection*{{Description over general schemes}}\label{DescriptionOverGeneralSchemes}

For $S$ a [[scheme]], a [[cubic curve]] over $S$ is a scheme $p \colon X \to S$ over $S$ equipped with a [[section]] $e \colon S \to X$ and such that [[Zariski topology|Zariski locally]] on $S$, $X$ is given by an [[equation]] in $\mathbb{P}_S^2$ of the form

\begin{displaymath}
y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6
\end{displaymath}
such that $e \colon S \to X$ is the line at infinity.

Euivalently this says that $p$ is a [[proper morphism|proper]] [[flat morphism]] with a section contained in the [[smooth locus]] whose [[fibers]] are geometrically integral curves of [[arithmetic genus]] one.

Write $\mathcal{M}_{cub}$ for the [[moduli stack]] of such [[cubic curves]]. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$

\begin{displaymath}
\mathcal{M}_{ell} \hookrightarrow \mathcal{M}_{cub} 
  \longrightarrow \mathcal{M}_{FG}.
\end{displaymath}
See at \emph{[[elliptic curve]]} for details. (A textbook account is in \hyperlink{Silverman09}{Silverman 09, III}, a review with an eye towards [[tmf]] is in \hyperlink{Mathew}{Mathew, section 3}).

Two standard versions of [[Hopf algebroids]] representing $\mathcal{M}_{\overline{ell}}$ as a [[geometric stack]] are usefully reviewed in (\hyperlink{Mathew}{Mathew, section 4}).

\hypertarget{InEInfinityGeometry}{}\subsection*{{As a derived scheme in $E_\infty$-geometry}}\label{InEInfinityGeometry}

By the [[Goerss-Hopkins-Miller theorem]] the [[structure sheaf]] $\mathcal{O}$ of the moduli stack of elliptic curves lifts to a sheaf $\mathcal{O}^{top}$ of [[E-∞ rings]] which over a given [[elliptic curve]] is the corresponding [[elliptic spectrum]].

By ([[A Survey of Elliptic Cohomology|Lurie (Survey), theorem 4.1]]), this yields a [[spectral Deligne-Mumford stack]] refinement

\begin{displaymath}
\mathcal{M}_{ell}^{der} \coloneqq  (\mathcal{M}_{ell}, \mathcal{O}^{top})
\end{displaymath}
which is the moduli stack of [[derived elliptic curves]], in that there is a [[natural equivalence]] in [[E-∞ rings]] $A$ of the form

\begin{displaymath}
Hom(Spec(A), \mathcal{M}_{ell}^{der})
  \simeq
  E(A)
  \,,
\end{displaymath}
where on the left we have maps of [[structured (∞,1)-toposes]] and on the right the [[∞-groupoid]] of [[derived elliptic curves]] over $A$.

This is based on the representability theorem ([[A Survey of Elliptic Cohomology|Lurie (Survey), prop. 4.1]], [[Representability Theorems|Lurie (Representability)]]).

In this derived picture the compactified dericed moduli space is obtained by gluing in the [[spectrum]] of [[Tate K-theory]] $KO[ [q] ] \simeq KU[ [q] ]/\mathbb{Z}_2$ by forming the [[homotopy pushout]]

\begin{displaymath}
\itexarray{
    Spec(K((q))) &\longrightarrow& \mathcal{M}_{ell}^{der} 
    \\
    \downarrow && \downarrow
    \\
    Spec(K[ [q] ]) &\longrightarrow& \mathcal{M}_{\overline{ell}}^{der}
  }
  \,.
\end{displaymath}
([[A Survey of Elliptic Cohomology|Lurie(Survey), p. 33]]). Again, the underlying ordinary [[Deligne-Mumford stack]] is the ordinary $\mathcal{M}_{\overline{\ell}}$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{finite_covers}{}\subsubsection*{{Finite covers}}\label{finite_covers}

The [[moduli space]] $\mathcal{M}_{ell}[n]$ of [[elliptic curves with level-n structure]] (for some $n \in \mathbb{N}$) provides a finite [[covering]] of $\mathcal{M}_{ell}$ (similarly for the compactifications).

(Over the [[complex numbers]] this is the \emph{[[modular curve]]}).

\hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology}

\begin{prop}
\label{}\hypertarget{}{}
\begin{displaymath}
H_1(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12\mathbb{Z}
\end{displaymath}
\begin{displaymath}
H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0
\end{displaymath}
\begin{displaymath}
H^2(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12 \mathbb{Z}
\end{displaymath}
\begin{displaymath}
H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q})
  \simeq
  H_\bullet(M_{1,1}, \mathbb{Q})
\end{displaymath}
and similarly for [[integral cohomology]]

\begin{displaymath}
\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
\end{displaymath}
\begin{displaymath}
Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}
\end{displaymath}
\end{prop}
\hypertarget{EulerCharacteristic}{}\subsubsection*{{Euler characteristic}}\label{EulerCharacteristic}

The [[orbifold Euler characteristic]] of the moduli space of complex elliptic curves is given by the [[special values of L-functions|special value]] of the [[Riemann zeta function]] at $s= -1$

\begin{displaymath}
\chi(\mathcal{M}_{1,1}) = \zeta(-1) = - \frac{1}{12}
  \,.
\end{displaymath}
This is a special case of the result in (\hyperlink{ZagierHarer86}{Zagier-Harer 86}) discussed at \emph{[[moduli space of curves]]}. See also the first page here: [[EulerCharacteristicOfSpaceOfCurves.pdf:file]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[modular form]]


\item [[elliptic cohomology]]



\end{itemize}
[[!include moduli spaces -- contents]]

[[!include moduli stack of curves -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Introductory lecture notes on the moduli space of elliptic curves over the [[complex numbers]] include

\begin{itemize}%
\item Richard Hain, \emph{Lectures on Moduli Spaces of Elliptic Curves} (\href{http://arxiv.org/abs/0812.1803}{arXiv:0812.1803})


\item section 4 of \emph{Introduction to Orbifolds} ([[IntroductionToOrbifolds.pdf:file]])



\end{itemize}
Accounts of the general case include

\begin{itemize}%
\item Nicholas M. Katz, [[Barry Mazur]], \emph{Arithmetic moduli of elliptic curves}, Annals of Mathematics Studies\_, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR MR772569 (86i:11024)


\item [[Joseph Silverman]], \emph{The arithmetic of elliptic curves}, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)



\end{itemize}
Lecture notes/talk notes reviewing this include

\begin{itemize}%
\item James Parson, \emph{Moduli of elliptic curves} (\href{http://math.stanford.edu/~conrad/vigregroup/vigre03/moduli.pdf}{pdf})


\item [[Akhil Mathew]], section 3 of \emph{The homotopy groups of $TMF$} (\href{http://math.mit.edu/~sglasman/tmfhomotopy.pdf}{pdf})


\item [[Andre Henriques]], \emph{The moduli stack of elliptic curves} (\href{http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter04/henriques.pdf}{pdf}) in \emph{Topological modular forms} Talbot workshop 2007 (\href{http://math.mit.edu/conferences/talbot/2007/tmfproc/}{web})



\end{itemize}
For more of the general picture in view of [[elliptic cohomology]] and [[tmf]] see also

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}

\end{itemize}
The [[orbifold Euler characteristic]] of the moduli space of curves was originally computed in

\begin{itemize}%
\item [[Don Zagier]], John Harer, \emph{The Euler characteristic of the moduli space of curves}, Inventiones mathematicae (1986) Volume: 85, page 457-486 (\href{https://eudml.org/doc/143377}{EUDML})

\end{itemize}
Reviews of the orbifold Euler characteristic computation include

\begin{itemize}%
\item \emph{Mathematical ideas and notions in quantum field theory -- 5. The Euler characteristic of the moduli space of curves} ([[EulerCharacteristicOfSpaceOfCurves.pdf:file]])

\end{itemize}
[[!redirects moduli space of elliptic curves]]



\end{document}