We provide an account of the construction of the moduli stack of elliptic curves as an analytic orbifold. While intimately linked to Thurston's point of view on the subject (discrete groups acting properly and effectively on differentiable manifolds), the construction of the modular orbi-curve and its universal family of elliptic curves ends up requiring a bit more technology, in order to allow for non-effective actions.

Primary 14D23; Secondary 32G13

§ INTRODUCTION

Orbifolds were first introduced by Satake, under the name $V$-manifolds ([11]), as a way to handle the ramification phenomena that occur in the general theory of group actions on manifolds. And it is generally accepted in the community that Satake chose that name in reference to the German word Verzweigung, which means ramification.[The second-named author believes he heard this explanation of the term in a talk by Professor Kaoru Ono, who asked Satake himself directly about it.] Now, ramification is a concept that is also present in the analytic or algebraic contexts, in particular to talk about morphisms that are not étale. And it is in fact remarkable that some of the concepts introduced by Satake and Thurston for differential-geometric purposes, namely orbifolds and their fundamental groups ([12, 16]), have natural algebro-geometric counterparts that can in turn be used to enrich our perspective and the scope of our methods in differential and analytic geometry. One of the goals of the present chapter is to illustrate this point of view through the example of the moduli space of elliptic curves.

It is well-known that elliptic curves can be classified by their $j$-invariant and this is the basis for the construction of the so-called modular curve, whose points parameterise isomorphism classes of elliptic curves ([9]). However, there are various reasons why we cannot quite content ourselves with the classical construction of this space. The main one, perhaps, being that the existence of a universal family of elliptic curves can only be obtained in the context of stacks, not manifolds. In this chapter, we will present a construction of the moduli stack of elliptic curves and show that it is in fact an analytic orbifold. This illustrates the need to go beyond the original definition of an orbifold because, in Thurston's definition of the quotient orbifold $[X/G]$, the group $G$ is supposed to act effectively on the manifold $X$, but to prove that the moduli stack of elliptic curves is analytic, one ends up describing it as the quotient stack $[\fh/\SL(2;\Z)]$, where $\fh$ is the upper half-plane in $\C$ and $\SL(2;\Z)$ acts on it by homographic transformations. This action is not effective, and in fact its kernel is precisely $\{\pm I_2\}$, so $\PSL(2;\Z)$ acts effectively on $\fh$. While $\fh/\SL(2;\Z)$ and $\fh/\PSL(2;\Z)$ are homeomorphic topological spaces, the quotient stacks $[\fh/\SL(2;\Z)]$ and $[\fh/\PSL(2;\Z)]$ are not isomorphic, and only the first one is isomorphic to the moduli stack of elliptic curves, so the fact that $\fh/\PSL(2;\Z)$ is an orbifold in the sense of Thurston is not entirely satisfying here, even though it does get us the correct coarse moduli space. Resolving this issue by showing that $\fh/\SL(2;\Z)$ carries a canonical structure of non-effective orbifold is not quite good enough either, because of the difficulties that arise with the universal family. So the only way out of this seems to be considering the quotient stack $[\fh/\SL(2;\Z)]$ and showing that it is an orbifold in the stacky sense. Of course, there are deeper implications to this discussion, for instance the fact that the fundamental group of the moduli space of elliptic curves should be $\SL(2;\Z)$, not $\PSL(2;\Z)$, because elliptic curves always have an Abelian group structure, so in particular an inversion map $x\lmt -x$, which is the map coming from the element $-I_2\in \SL(2;\Z)$ acting trivially on $\fh$.

The example of the moduli space of elliptic curves illustrates how thinking about non-effective orbifolds as a particular class of stacks may be useful, even for classical problems. This suggests the need to go beyond Satake and Thurston's original definitions, using the input from algebraic geometry to enrich the differential-geometric perspective on the notion of orbifold. Note that this is not the only occurrence of such a phenomenon. While no evidence suggests that the theory of the étale fundamental group of schemes may have been of inspiration to Thurston (except perhaps his construction of the universal cover as an inverse limit in <cit.>), the analogy is clear, and in fact illustrates a common trend between the two fields, which is to include more maps in the definition of a covering space in order to have a richer notion of fundamental group. Indeed, consider for instance the orthogonal projection from the unit sphere in $\R^3$ to the closed unit disk in $\R^2\times\{0\}\subset\R^3$. In the usual topology of $\R^3$, this map is not a covering map (in fact, it is not even locally injective at points of the unit circle). But in algebraic geometry, it is the map induced on the set of closed points by the canonical projection $\P^1_\C\to\P^1_\R=\P^1_\C/\Gal(\C/\R)$, which is an étale morphism of schemes (in particular, the étale fundamental group of $\P^1_\R$ is not trivial, and since $\P^1_\C$ is simply connected, it is in fact exactly $\Gal(\C/\R)$). If we view this example through the orbifold prism, the orbifold fundamental group, in the sense of Thurston, of the quotient of $S^2\subset\R^3$ by the group generated by the reflection through the equator is indeed $\Z/2\Z$, just like in algebraic geometry. More generally, this leads to a richer notion of fundamental groups for all Klein surfaces (= quotients of Riemann surfaces by anti-holomorphic involutions), for which we refer for instance to [13]. In analytic and algebraic geometry too, it is useful to allow orbifolds: the fundamental group of the quotient of $\C$ by the finite group $\mu_n(\C)\simeq\Z/n\Z$ acting by multiplication by a primitive $n$-th root of unity should be $\mu_n(\C)$, not the trivial group. As a matter of fact, the canonical projection $\C\lra \C/\mu_n(\C)$ is the typical case of a map which is ramified when seen as a morphism between smooth algebraic or analytic varieties (in which case it can be identified with the map $z\lmt z^n$), but which can be viewed as a covering map in the orbifold sense (see Example <ref>).

Recall that a complex-analytic elliptic curve is a pair $(\sC,e)$ where $\sC$ is a compact Riemann surface of genus $1$ and $e$ is a point in $\sC$. A morphism between two elliptic curves $(\sC_1,e_1)$ and $(\sC_2,e_2)$ is a holomorphic map $f:\sC_1\lra \sC_2$ such that $f(e_1)=e_2$. We shall denote by $f:(\sC_1,e_1)\lra (\sC_2,e_2)$ such a morphism. The main result that we wish to discuss in this chapter is stated as follows.

[Theorem <ref>]
The moduli stack of complex elliptic curves is a complex analytic stack, isomorphic to the orbifold $[\fh/\SL(2,\Z)]$, where $\fh:=\{z\in\C\ |\ \mathrm{Im}\,(z)>0\}$ is the hyperbolic plane and
$$\SL(2,\Z):=\left\{\begin{pmatrix} a & c \\ b & d \end{pmatrix} : a,b,c,d \in \Z,\, ad-bc=1\right\}$$
is the so-called modular group, acting to the right on $\mathfrak{h}$ by homographic transformations

$$ z \cdot \begin{pmatrix}a & c \\ b & d \end{pmatrix} = \frac{az+b}{cz+d}%\quad \left(\text{or equivalently, } A\cdot z := A^t \begin{pmatrix} z \\ 1 \end{pmatrix} \, \right)
\, .$$

This moduli stack admits the (non-compact) Riemann surface $\fh/\SL(2,\Z) \simeq \C$ as a coarse moduli space.

In Section <ref>, we recall basic results on elliptic curves. Sections <ref> and <ref> contain the formal definition of a stack and an explanation of what it means for a stack defined on the category of complex analytic manifolds to be analytic, and to be an orbifold. In Sections <ref> and <ref>, we discuss the notion of moduli spaces for a  complex analytic stack and we prove that the stack of elliptic curves is an analytic orbifold.

This chapter is entirely expository and makes no claims to originality: its main goal is to be self-contained enough in order to be useful to young researchers who are entering the field and are interested in the interactions between differential and algebraic geometry, all the while celebrating the vitality and scope of Thurston's geometric vision.

§ COMPLEX ELLIPTIC CURVES

§.§ Complex tori

Let us start by recalling the definition of a complex elliptic curve (in the analytic setting). We recall that the (geometric) genus of a compact connected Riemann surface $\sC$ is the dimension of the complex vector space $H^{0}(\sC;\Omega^{1}_{\sC})$ which, as $\sC$ is $1$-dimensional and compact, is the space of holomorphic $1$-forms on $\sC$.

An analytic complex elliptic curveelliptic curve is a pair $(\sC,e)$ where $\sC$ is a compact Riemann surface of genus $1$ and $e$ is a point in $\sC$. A morphism of elliptic curves between $(\sC,e)$ and $(\sC',e')$ is a holomorphic map $f: \sC \lra \sC'$ such that $f(e)=e'$.

We now give the two sources of examples of complex elliptic curves that will be of interest to us: complex tori (Example <ref>) and smooth plane cubics (Example <ref>).

Let us see $\C \simeq \R^2$ as a two-dimensional real vector space (equipped with its usual topology and additive group structure) and let $\Lambda$ be a discrete subgroup of $\C$ such that the quotient group $\C/\Lambda$ is compact (such a subgroup $\Lambda \subset \C$ will be called a (uniform) latticelattice). In particular, the group $\Lambda$ is a free $\Z$-module of rank $2$. Note that there is a canonical marked point in $\C/\Lambda$ in this case, namely $0\ \text{mod}\ \Lambda$, the neutral element for the group law. Moreover, there is a (unique) Riemann surface structure on $\C / \Lambda$ that makes the canonical projection $p: \C \lra \C/\Lambda$ holomorphic. A holomorphic $1$-form on $\C/\Lambda$ is necessarily closed, so it pulls back to an exact global $1$-form $f(z)\, dz$ on the simply connected Riemann surface $ \C $, with $f(z)$ holomorphic and $\Lambda$-periodic (in the sense that, if $ z \in \C $ and $\lambda \in \Lambda$, then $f(z+\lambda) = f(z)$; this holds because $dz$ is $\Lambda$-invariant for the action of $\Lambda$ on $\C$ by translation). Since $\C/\Lambda$ is compact, the holomorphic function $f$ has to be bounded, therefore constant by Liouville's theorem, and this proves that the space of holomorphic $1-$forms on $\mathbb{C}/\Lambda$ has complex dimension 1, hence that we have constructed a complex elliptic curve $(\C/\Lambda, 0\ \text{mod}\ \Lambda)$, which we will refer to as a ($1$-dimensional) complex toruscomplex torus.

The complex tori of Example <ref> can be embedded onto non-singular complex projective sets using the Weierstrass $\wp$-function of the lattice $\Lambda$, namely

\wp_\Lambda(w) = \frac{1}{w^2} + \sum_{\lambda \in \Lambda\setminus\{0\}} \left( \frac{1}{(w-\lambda)^2} - \frac{1}{\lambda^2} \right)

which, if we pick a basis $(\lambda,\mu)$ of $\Lambda$ as a $\Z$-module, satisfies the equation

\begin{equation}\label{image_proj_embedding}
(\wp_\Lambda'(w))^2 = f_{(\lambda,\mu)}\big(\wp_\Lambda(w)\big),
\end{equation}

f_{(\lambda,\mu)}(x) = 4 (x - v_1) (x-v_2) (x-v_3)$$

v_1 = \wp_\Lambda\left(\textstyle\frac{\lambda}{2}\right) \in \C,  v_2 = \wp_\Lambda\left(\textstyle\frac{\mu}{2}\right) \in \C \ \text{and} \ v_3 = \wp_\Lambda\left(\textstyle\frac{\lambda+\mu}{2}\right) \in \C\, ,

thus inducing a holomorphic map

\begin{equation}\label{embedding_cx_torus_in_proj_plane}
\begin{array}{rcl}
\C/\Lambda & \lra & \CP^2 \\
w \text{ mod } \Lambda & \lmt & \left[ \wp_\Lambda(w) : \wp_\Lambda'(w) : 1 \right]
\end{array}
\end{equation}

whose image, because of Equation (<ref>) and up to normalisation of the basis $(\lambda,\mu)$, is a smooth projective curve of the type described in Equation (<ref>).

Let $u\in \C\setminus\{0,1\}$ and consider the smooth projective curve

\begin{equation}\label{eq123}
\sC_{u} := \left\{ [x:y:z] \in\mathbb{CP}^{2} \ | \ y^{2}z=x(x-z)(x-u z)\right\}.
\end{equation}

By the genus-degree formula, the smooth cubic $\sC_{u}$ has genus $ g = \frac{ ( 3 - 1 ) ( 3 - 2 ) }{ 2 } = 1 $. And since the point $e:=[0:0:1]$  belongs to $\sC_{u}$ for all $u\in\mathbb{C}\setminus\{0,1\}$, we have defined a family of elliptic curves $(\sC_{u},e)_{u\in\C\setminus\{0,1\}}$, depending on the parameter $u$. This family is called the Legendre familyLegendre family of elliptic curves (see also Example <ref>).

Building on Example <ref>, let us set $f_u(x) = x(x-1)(x-u)$. The classical theory of elliptic curves says that $\sC_u$ is a compactification of the plane cubic of equation $y^2 = f_u(x)$. Moreover, the map $(x,y) \lmt x$ induces a ramified double covering $p: \sC_u \lra \CP^1$, with four branch points, so an application of the Riemann-Hurwitz formula shows that $\sC_u$ is homeomorphic to $S^1 \times S^1$. If we pick a square root of $ f_u $, the differential form $\frac{dt}{\sqrt{f_u(t)}}$ on $\CP^1$ pulls back to a holomorphic $1$-form  $\omega_u := p^*\frac{dt}{\sqrt{f_u(t)}}$ that can be extended holomorphically to $\sC_u$. Then we can integrate this $1$-form along loops based at $e \in \sC_u$ and, since $\omega_u$ is closed, obtain a group morphism

\begin{equation}\label{period_subgroup}
\Phi_{\omega_u}:
\begin{array}{rcl}
\pi_1(\sC_u, e) & \lra & \C \\
\left[\gamma\right] & \lmt & \int_\gamma \omega_u
\end{array}
\end{equation}

whose image $\Lambda_{\omega_u} := \mathrm{Im}\,(\Phi_{\omega_u})$ is in fact a lattice in $\C$, called the period latticelattice!period lattice. This induces a holomorphic map

\begin{equation}\label{isom_smooth_plane_cubic_cx_torus}
\begin{array}{rcl}
\sC_u & \lra & \C/\Lambda_{\omega_u} \\
x & \lmt & \int_{e}^{x} \omega_u \ \text{mod} \ \Lambda_{\omega_u}
\end{array}
\end{equation}

which is an isomorphism of elliptic curves $(\sC_u,e) \simeq (\C/\Lambda_{\omega_u}, 0\ \text{mod}\  \Lambda_{\omega_u})$. Note that if we choose a different non-zero holomorphic $1$-form $\omega$ on $\sC_u$, it will satisfy $\omega = \alpha \omega_u$ for some $\alpha\in\C^*$, because the space of such forms is $1$-dimensional. Consequently, the lattices $\Lambda_{\omega_u}$ and $\Lambda_{\omega}$ satisfy  $\Lambda_{\omega} = \alpha  \Lambda_{\omega_u}$, so multiplication by $\alpha$ in $\C$ induces an isomorphism of elliptic curves $(\C/\Lambda_{\omega_u}, 0\ \text{mod}\ \Lambda_{\omega_u}) \simeq (\C/\Lambda_\omega, 0\ \text{mod}\ \Lambda_\omega)$.

So, from the existence of the projective embedding (<ref>) and the isomorphism (<ref>), we see that there is a bijection

\big\{ 1\text{-dimensional complex tori} \big\} \, \big/ \, \text{isomorphism} \simeq \big\{ \text{smooth plane cubics} \big\} \, \big/ \, \text{isomorphism} \, .

And as a matter of fact, every elliptic curve $(\sC,e)$ in the sense of Definition <ref> is of this form, meaning that it is isomorphic to a one-dimensional complex torus. One way to show this is to generalise the construction of the period subgroup (<ref>) to the abstract setting: choose a generator $\omega$ of the $1$-dimensional complex vector space $H^0(\sC;\Omega^{1}_{\sC})$ and show that the image of the group morphism $\Phi_\omega$ is a lattice $\Lambda_\omega \subset \C$ and that there is an isomorphism of elliptic curves $(\sC,e) \simeq (\C/\Lambda_\omega, 0 \ \text{mod}\ \Lambda_\omega)$. The issue here is that, with our definition of the genus, we do not yet know that $\sC$ is homeomorphic to $S^1\times S^1$, which makes it harder to prove that the period subgroup is a lattice in $\C$. The way to solve this difficulty is to use Abel's theorem, which gives a necessary and sufficient condition for a Weil divisor $D$ on a compact Riemann surface $\sC$ to be  a principal divisor, i.e. for $D$ to be the divisor of zeros and poles of a meromorphic function on $\sC$. More precisely, one can follow the approach in <cit.> to show the following:

* A Weil divisor $D\in \Div(\sC)$ is the divisor of zeros and poles of a meromorphic function $f$ on $\sC$ if and only if $D$ is of degree $0$ and there exists a chain $c \in C_1(\sC;\Z)$ such that $\partial c = D$ and $\int_c \omega = 0$ (Abel's theorem in the $g=1$ case, <cit.>).

* The period subgroup $\Lambda_\omega := \mathrm{Im}\,\Phi_\omega \subset \C$ is a lattice <cit.>. This uses the previously stated version of Abel's theorem, to prove that $\Lambda_\omega$ is a discrete subgroup of $\C$.

One gets in this way a $1$-dimensional complex torus $\mathrm{Jac}(\sC) := \C / \Lambda_\omega$, namely the Jacobian variety of the elliptic curve $(\sC,e)$. It depends on $\omega$, but if we choose a different basis $\alpha \omega$ of the complex vector space $H^0(\sC;\Omega^{1}_{\sC})$, we obtain an isomorphic complex torus, as we have already seen. Moreover, the Riemann-Roch theorem has the following consequence.

Let $(\sC,e)$ be an elliptic curve and let $\mathrm{Pic}_0(\sC) := \Div_0(\sC) / \Div_P(\sC)$ be the group of degree $0$ divisors modulo principal divisors. Then the map

\Psi:
\begin{array}{rcl}
\sC  & \lra & \mathrm{Pic}_0(\sC) \\
x & \lmt & [x] - [e]
\end{array}

is bijective.

The map $\Psi$ is injective because, if $\Psi(x) = \Psi(x')$ with $x\neq x'$, then the Weil divisor $[x]-[x']$ would be principal, so there would exist a meromorphic function $f$ on $\sC$ with exactly one pole, of order $1$, and the corresponding holomorphic map $f:\sC \lra \CP^1$ would be an isomorphism, contradicting the fact that $\dim_\C H^0(\sC;\Omega^{1}_{\sC}) = 1$.
Moreover, the map $\Psi$ is surjective because every divisor $D$ of degree $0$ on $\sC$ is equivalent, modulo a principal divisor, to a divisor of the form $[x]-[e]$. Indeed, if we set $D' = D +[e]$, then $\deg(D') = \deg(D)+1= 1$ so, by the Riemann-Roch theorem, $\dim H^0(\sC;O_\sC(D')) \geq \deg(D') = 1$, meaning that there exists a non-zero meromorphic function $f$ on $\sC$ whose poles are bounded by $D'$, i.e. in terms of divisors, $(f) + D' \geq 0$. Since $(f)+D'$ is an effective Weil divisor of degree $1$, there exists a point $x$ in $\sC$ such that $(f)+D' = [x]$, hence $D = D' - [e] \sim_{(f)} [x] - [e]$.

Lemma <ref> shows in particular that an elliptic curve $(\sC,e)$ has a natural structure of Abelian group, induced by the isomorphism $\Psi: (\sC,e) \overset{\simeq}{\lra} (\mathrm{Pic}_0(\sC),0)$ and the group structure on $\mathrm{Pic}_0(\sC)$. In particular, an elliptic curve always has non-trivial automorphisms (namely the inversion map $D \lmt -D$, seen as a morphism of marked Riemann surfaces).

Finally, one can deduce from the above that a complex elliptic curve $(\sC,e)$ is isomorphic to a complex torus, namely the Jacobian variety $\mathrm{Jac}(\sC) := \C/\Lambda_\omega$, defined by the period lattice. This proves in particular that $\pi_1(\sC,e) \simeq \Lambda_\omega \simeq \Z^2$. So we see from the classification of compact connected orientable surfaces that $\sC$ is homeomorphic to $S^1\times S^1$.

Let $(\sC,e)$ be a complex elliptic curve and let $\omega$ be a non-zero holomorphic $1$-form on $\sC$. Let $\Lambda_\omega \subset \C$ be the period lattice and let $\mathrm{Jac}(\sC) := \C/\Lambda_\omega$ be the associated complex torus, i.e. the Jacobian variety of $(\sC,e)$. Then the map

\begin{array}{rcl}
\sC & \lra & \mathrm{Jac}(\sC) = \C/\Lambda_{\omega} \\
x & \lmt & \int_{e}^{x} \omega \ \mathrm{mod} \ \Lambda_{\omega}
\end{array}
is an isomorphism of elliptic curves.

We refer to <cit.> for a complete proof of Theorem <ref>. Note that the map $F:\sC \lra \mathrm{Jac}(\sC)$ is the composition of the map $\Psi: \sC \lra  \mathrm{Pic}_0(\sC)$ of Lemma <ref> with the map

\begin{array}{rcl}
\mathrm{Pic}_0(\sC) & \lra & \mathrm{Jac}(\sC) \\
D = \partial c & \lmt & \int_{c} \omega \ \text{mod} \ \Lambda_{\omega}
\end{array}
which is well-defined by definition of the period lattice (i.e. independent of the choice of the chain $c\in C_1(\sC;\Z)$ such that $\partial c = D$), and injective in view of Abel's theorem. The surjectivity of $J$ is proven in <cit.> and for the fact that the map $F:\sC \lra  \mathrm{Jac}(\sC)$ is holomorphic, we refer to <cit.>.

As an application, Theorem <ref> gives us the structure of the automorphism group of a complex elliptic curve and of a compact Riemann surface of genus $1$ without marked point. To obtain it, one just needs to observe that two complex tori $\C/\Lambda_1$ and $\C/\Lambda_2$ are isomorphic as complex elliptic curves if and only if there exists $\alpha\in\C^*$ such that $\alpha\Lambda_1 = \Lambda_2$, which can be deduced by lifting such an isomorphism to the universal cover $\C$ of these tori and using the classification of automorphisms of $\C$. In particular, the automorphism group of the elliptic curve $(\C/\Lambda,0)$ can be identified with the group of all $\alpha\in\C^*$ such that $\alpha\Lambda = \Lambda$. As $\Lambda$ is a discrete subgroup of $\C$, the set of such $\alpha$ is in fact a discrete subgroup of the circle group $S^1$, hence a finite rotation group, generated by a root of unity. If we do not ask to preserve the base point, then translations by an element $[z]\in\C/\Lambda$ also induce automorphisms of $\C/\Lambda$.

Let $(\sC,e)$ be a complex elliptic curve. Then there exists a natural number $n\geq 1$ such that the automorphism group of the marked Riemann surface $(\sC,e)$ is

\begin{equation}\label{autom_of_elliptic_curves}
\Aut(\sC,e) \simeq \Z/n\Z.
\end{equation}
Moreover, the automorphism group of the Riemann surface $\sC$ is the semi-direct product
$$\Aut(\sC) \simeq \sC \rtimes \Aut(\sC,e)$$
where the group $\sC\simeq\mathrm{Pic}_0(\sC)$ acts on itself by translations and the structure of semi-direct product is given by the action of $\Aut(\sC,e)$ on $\sC$. In particular, the group $\Aut(\sC)$ is a $1$-dimensional compact analytic Lie group over $\C$, whose identity component is isomorphic to $\sC$.

We will see in Corollary <ref> that, in fact, the natural number $n$ in Equation (<ref>) can only be $2$, $4$ or $6$ (never $1$!). In particular, we will give an alternate, more detailed proof of Corollary <ref>.

So the conclusion we can draw from the present paragraph is two-folded: firstly, the classification of complex elliptic curves up to isomorphism can be reduced to the classification of complex tori, and secondly, such objects have non-trivial automorphisms as marked Riemann surfaces. As a consequence, even though we can always find, given a complex elliptic curve $(\sC,e)$, an isomorphism of elliptic curves with a complex torus $\C/\Lambda$, this isomorphism will never be unique, because we can compose it with an automorphism of $\C/\Lambda$. This is one of the reasons why constructing a moduli space of complex elliptic curves requires additional notions, namely framings, in order to get rid of non-trivial automorphisms.

§.§ Framed elliptic curves

When faced with a moduli problem for objects that have non-trivial automorphisms, an idea that has proven useful is to add an extra structure to these objects. This is because the corresponding notion of isomorphism will then be more restrictive (since the extra structure needs to be preserved, too), so in particular there will be fewer automorphisms. In the best case scenario, there is a group action on these objects with extra structure, and two such objects with extra structure lie in the same orbit if and only if they are isomorphic as objects without extra structure, so the moduli space of objects without extra structure can be constructed as a quotient of the moduli space of objects with extra structure (provided the latter moduli space actually exists, of course). We will now illustrate this more concretely in the case of elliptic curves. We begin with the definition of an extra structure.

elliptic curve!framed elliptic curveframing
Let $(\sC,e)$ be a complex elliptic curve. A framing of $(\sC,e)$ is a basis $([a],[b])$ of the free $\Z$-module $H_{1}(\sC,\mathbb{Z}) \simeq \pi_1(\sC,e) \simeq \Z^2$, with the additional property that

\begin{equation}\label{positivity_cond}
\text{for all non-zero } \omega \in H^0\big(\sC;\Omega^1_{\sC}\big),\ \mathrm{Im}\, \textstyle \frac{\int_a \omega}{\int_b \omega} >0\,.
\end{equation}
Since $H^0(\sC;\Omega^1_{\sC})$ is one-dimensional by assumption, it suffices to test Condition (<ref>) on a single non-zero holomorphic $1$-form $\omega$. A morphism of framed elliptic curves
$$f: \big(\sC,e,([a],[b])\big) \lra \big(\sC',e',([a'],[b'])\big)$$
is a morphism of elliptic curves $f:(\sC,e) \lra (\sC',e')$ such that $f_*([a]) = [a']$ and $f_*[b] = [b']$, where $f_*: H_1(\sC;\Z) \lra H_1(\sC';\Z)$ is the group morphism induced by $f$ in homology.

If $(\sC,e) = (\C/\Lambda,0)$ for some lattice $\Lambda \subset \C$ and $(\lambda,\mu)$ is  a basis  of the $\Z$-module $\Lambda$ that satisfies the additional positivity condition $\mathrm{Im}\,\frac{\mu}{\lambda} >0$, then $(\lambda,\mu)$ defines a framing of $(\C/\Lambda,0)$ by letting $\hat{\lambda}$ and $\hat{\mu}$ be the $1$-cycles in $\C/\Lambda$ corresponding to the paths $t \lmt t \lambda$ and $t \lmt t \mu$ in $\C$. A lattice $\Lambda\subset \C$ equipped with such a direct basis $(\lambda,\mu)$ is also called a framed latticelattice!framed lattice, and we shall sometimes identify a framing of $\Lambda$ in this sense with the induced framing of $\C/\Lambda$. So, when we write $(\C/\Lambda,0,(\lambda,\mu))$, we mean the framed elliptic curve $(\C/\Lambda,0,(\hat{\lambda},\hat{\mu}))$. Note that if a non-zero holomorphic $1$-form $\omega$ on $\sC$ is fixed, then we can define $(\lambda,\mu)$ from $([a],[b])$ by setting $\lambda:=\int_a \omega$ and $\mu:=\int_b \omega$, but without such an $\omega$, we cannot recover $(\lambda,\mu)$ from $(\hat{\lambda},\hat{\mu})$. Regardless, if $\Lambda' = \alpha\Lambda$ for some $\alpha\in\C^*$, then the map $z \lmt \alpha z$ induces an isomorphism of framed elliptic curves

\big(\C/\Lambda,0,(\lambda,\mu)\big) \simeq \big(\C/(\alpha\Lambda),0,(\alpha\lambda,\alpha\mu)\big).
In particular, if we set $\tau := \frac{\mu}{\lambda}$, then $\tau$ is a complex number with positive imaginary part, i.e. $\mathrm{Im}\,(\tau) >0$, and there is an associated lattice $\Lambda(\tau):= \Z1 \oplus \Z \tau \subset \C$ with direct basis $(1,\tau)$ such that the framed elliptic curve $(\C/\Lambda, 0, (\lambda,\mu))$ is isomorphic to the framed elliptic curve $(\C/\Lambda(\tau), 0, (1,\tau))$.

In view of Theorem <ref> and Example <ref>, a framed complex elliptic curve $(\sC,e,([a],[b]))$ is isomorphic to a framed complex torus of the form $(\C/\Lambda(\tau), 0, (1,\tau))$ where
$$ \tau \in \fh := \{z\in\C \ |\ \mathrm{Im}\,(z) >0\}.$$
Indeed, it suffices to show that a complex elliptic curve admits a framing, which comes from the simple observation that if a basis $([a],[b])$ for $H_1(\sC;\Z)$ does not satisfy the positivity condition (<ref>), then the basis $([b],[a])$ does. Note that, a priori, given a framed elliptic curve $(\sC,e,([a],[b]))$, there could be more than one $\tau\in \fh$ such that $(\sC,e,([a],[b])) \simeq (\C/\Lambda(\tau), 0, (1,\tau))$. But Lemma <ref> shows that this is in fact not the case. In particular, a framed elliptic curve of the form $(\C/\Lambda(\tau), 0, (1,\tau))$ has no non-trivial automorphisms.

The framed complex tori $(\C/\Lambda(\tau), 0, (1,\tau))$ and $(\C/\Lambda(\tau'), 0, (1,\tau'))$ are isomorphic if and only if $\tau=\tau'$.

An isomorphism between such complex tori is induced by $z \lmt \alpha z$ for some $\alpha\in\C^*$, which respects the framing if and only if $\alpha=1$ and $\tau'=\tau$.

It is not entirely apparent from the proof, but the positivity condition $\mathrm{Im}\,(\tau) >0$ is necessary if we want a uniqueness result such as the one in Lemma <ref> to hold. Indeed, if we had defined a framing of $\Lambda$ simply as a basis of the $\Z$-module $\Lambda$, then, even if we normalise the lattice so that the first vector in the basis be equal to $1$, two normalised framings of the complex torus $(\C/\Lambda(\tau), 0)$ do not have to be equal: of the two bases $(1,\tau)$ and $(\tau,1) \sim \big(1,\frac{1}{\tau}\big)$ of the lattice $\Lambda(\tau) = \Z1\oplus\Z\tau = \Z\tau\oplus\Z 1$, only the first one satisfies the required positivity condition.

We then have the following classification result for framed elliptic curves, whose proof follows directly from the discussion above and Lemma <ref>.

Let $(\sC,e,([a],[b]))$ be a framed elliptic curve. Then there exists a unique

\tau\in \fh = \big\{ z \in \C \ | \ \mathrm{Im}\,(\tau)>0 \big\}

and a unique isomorphism of framed elliptic curves

$$ \big(\sC,e,([a],[b])\big) \simeq \big(\C/\Lambda(\tau), 0, (1,\tau)\big). $$
In particular, framed elliptic curves have no non-trivial automorphisms:
$$\Aut(\sC,e,([a],[b])) = \{\id_{\sC}\}.$$

As a consequence, the points of the Riemann surface

\fh := \{z\in\C \ |\ \mathrm{Im}\,(z) >0\}
represent isomorphism classes of framed elliptic curves. This parameter space of framed elliptic curves is called the Teichmüller spaceTeichmüller space of elliptic curves of elliptic curves, for reasons that will be explained next. Note that Theorem <ref> constitutes an improvement on Theorem <ref>, since we can now compare two given elliptic curves in a more formal sense because (at least when they are framed) these elliptic curves are represented by points of a same space, namely the upper half-plane $\fh$. The next step is to ask ourselves: when are two framed elliptic curves isomorphic as plain elliptic curves? This corresponds to forgetting the framing, so answering this question should also give us a sense of what the possible framings are on a given elliptic curve. More precisely, we will construct the set of isomorphism classes of elliptic curves as a quotient of the Teichmüller space $\fh$, so it will come equipped with a map from $\fh$ to it, and the fibres of that map will be in bijection with the possible framings (the map in question being the one that forgets the frame on a given framed elliptic curve).

Let $(\sC,e,([a],[b]))$ and $(\sC',e',([a'],[b']))$ be framed elliptic curves and let $\tau, \tau' \in \fh$ be the corresponding points in the Teichmüller space of elliptic curves. Then the unframed elliptic curves $(\sC,e)$ and $(\sC',e')$ are isomorphic as elliptic curves if and only if there exists a matrix
$$A=\begin{pmatrix} a & c \\ b & d \end{pmatrix} \in \SL(2;\Z) \text{ such that } \tau' = \frac{a\tau + b}{c\tau +d}\, ,$$
meaning that $\tau$ and $\tau'$ are related by a direct Möbius transformation with integral coefficients.

In view of Theorem <ref>, the framed elliptic curves $(\sC,e,([a],[b]))$ and $(\sC',e',([a'],[b']))$ can be replaced by the framed elliptic curves $(\C/\Lambda(\tau),0,(1,\tau))$ and $(\C/\Lambda(\tau'),0,(1,\tau'))$ and the latter are isomorphic as unframed elliptic curves if and only if there exists $\alpha\in\C^*$ such that $\alpha \Lambda(\tau) = \Lambda(\tau')=\Z1\oplus\Z\tau'$. So there exist $a,b,c,d\in\Z$ such that $\tau' = \alpha(a\tau + b)$ and $1 = \alpha(c\tau +d)$. This implies that

$$\alpha= \frac{1}{c\tau+d} \quad \text{and} \quad \tau' = \frac{a\tau +b}{c\tau+d}\,.$$
Applying the same argument to $ \Lambda(\tau)= \frac{1}{\alpha} \Lambda(\tau') $, we see that the (Möbius) transformation $z \lmt \frac{az + b}{cz+d}$ is bijective, so the matrix $\begin{pmatrix} a & c \\ b & d \end{pmatrix}$ is invertible, therefore has determinant $\pm1\in\Z$. And since it sends $\tau$  such that $\mathrm{Im}\,(\tau)>0$ to $\tau'$ such that $\mathrm{Im}\,(\tau')>0$, it must have determinant $+1$.

So the framed elliptic curves $(\sC,e,([a],[b]))$ and $(\sC',e',([a'],[b']))$ become isomorphic when forgetting the frame if and only if the points $\tau, \tau' \in \fh$ lie in a same orbit of the $\SL(2;\Z)$-action on $\fh$ defined by

\begin{equation}\label{action_mod_group}
z \cdot \begin{pmatrix}a & c \\ b & d \end{pmatrix} = \frac{az+b}{cz+d}\,.
\end{equation}

Note that this is a right action. Thanks to this, we can construct a set whose points are in bijection with
isomorphic classes of elliptic curves.

The quotient set $\fh/\SL(2;\Z)$ is the set of isomorphism classes of elliptic curves and we have a forgetful map

F: \fh \lra \fh/\SL(2;\Z)

sending the isomorphism class of the framed elliptic curve $(\C/\Lambda(\tau),0,(1,\tau))$ to the isomorphism class of the elliptic curve $(\C/\Lambda(\tau),0)$. In particular, the orbit of a point $\tau\in\fh$ under the $\SL(2;\Z)$-action (<ref>) is in bijection with the set of all framings on the elliptic curve $(\C/\Lambda(\tau),0)$, and the stabiliser group of $\tau$ in $\SL(2;\Z)$ is isomorphic to the automorphism group of the elliptic curve $(\C/\Lambda(\tau),0)$.

§.§ The modular curve
modular!modular curve

The modular curve is the quotient $\fh/\SL(2;\Z)$, of the Teichmüller space $\fh$ by the action of the modular groupmodular!modular group $\SL(2;\Z)$. Note that Corollary <ref> tells us that this quotient set should be the underlying set of the moduli space of elliptic curves.

In this section, we recall what the canonical structure of Riemann surface on $\fh/\SL(2;\Z)$ is. But first, the mere fact that it is a quotient set already gives us insight into the classification of complex elliptic curves: by studying the action of the modular group on the Teichmüller space (stabilisers and fundamental domain), we can determine what the possible automorphism groups of complex elliptic curves are, and what the quotient may be as a Riemann surface, which turns out to be intimately related to the structure of the modular group itself. Note for instance that the centre of the modular group is the subgroup $\{\pm I_2\} \simeq \Z/2\Z$ and that this subgroup acts trivially on the Teichmüller space. So every elliptic curve admits an order two automorphism. This is already apparent in Theorem <ref>, which endows every Riemann surface of genus $1$ with a marked point with a (non-canonical) structure of Abelian group: the associated order-two automorphism is just the inversion map $D\lmt (-D)$ in this Abelian group. And if the elliptic curve is given by a smooth cubic equation $y^2z = f(x,z)$ in the projective plane, then the involution is induced by the involution $[x:y:z] \lmt [x:(-y):z]$ of $\CP^2$. This also shows that the action of $\SL(2;\Z)$ on $\fh$ is not faithful, as the centre of the modular group acts trivially on $\fh$. Hence an induced action of $\PSL(2;\Z) := \SL(2;\Z) / \{\pm I_2\}$, which turns out to be faithful. The important observation here is that the orbits of the $\PSL(2;\Z)$-action are the same as those of the $\SL(2;\Z)$-action but the stabilisers are not the same. In particular, the $\PSL(2;\Z)$-stabiliser of a point $\tau\in\fh$ is not isomorphic to the automorphism group of the elliptic curve $(\C/\Lambda(\tau),0)$. The following result is classical (see for instance <cit.> or [3] for an exposition).

The modular group $\SL(2;\Z)$ is generated by the matrices

S = \left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)
\quad \text{and} \quad T
=  \left(\begin{array}{rr} 1 & 1 \\ 0 & 1  \end{array}\right)
\, ,
hence also by the matrices $S$ and $ R:=ST =  \left( \begin{array}{rr} 0 & -1 \\ 1 & 1 \end{array} \right) $. This induces the following presentations of $\SL(2;\Z)$ and $\PSL(2;\Z)$ by generators and relations:

$$ \SL(2;\Z) \ \simeq \ \big\langle \, S, R \ \big| \  S^2 = R^3,\ S^4 = R^6 = 1\, \big\rangle \  \simeq \ \Z/4\Z \ast_{\Z/2\Z} \Z/6\Z$$

$$ \PSL(2;\Z) \ \simeq \ \big\langle \, [S], [R] \ \big| \ [S]^2 = [R]^3 = 1 \big\rangle \ \simeq \ \Z/2\Z \ast \Z/3\Z.$$

Theorem <ref> is about the algebraic structure of $\SL(2;\Z)$ and $\PSL(2;\Z)$. The matrices $S$ and $R$ show that this group is generated by an element of order four and an element of order six, which themselves satisfy the relation $S^2=R^3$ (equal to $-I_2$). So $\SL(2;\Z)$ is the amalgamated coproduct of the two Abelian groups $\Z/4\Z$ and $\Z/6\Z$ over $\Z/2\Z$, with the non-trivial element of $\Z/2\Z$ being sent to $S^2$ in $\langle S \rangle \simeq \Z/4\Z$ and to $R^3$ in $ \langle R \rangle \simeq \Z/6\Z$. As a consequence, the group $\PSL(2;\Z) = \SL(2;\Z) / \{\pm I_2\}$ is generated by the classes $[S]$ and $[R]$, which are respectively of order $2$ and $3$ with no other relations between them, so $\PSL(2;\Z)$ is the coproduct of $\Z/2\Z$ and $\Z/3\Z$.  This implies for instance that a character $\chi:\SL(2;\Z)\lra \C^*$ has image contained in the subgroup generated by a fourth root of unity and a sixth root of unity, so $\mathrm{Im}\, \chi \subset \mu_{12}(\C)$. Geometrically, though, it is the action of the generators $S$ and $T$  that is the most convenient to visualise. Indeed, $S$ acts on $\fh$ via the Möbius transformation $[S]: z \lmt -\textstyle\frac{1}{z}$, which is a direct rotation of angle $\frac{\pi}{2}$ about $i$, and $T$ acts via $[T]: z \lmt z+1$, which is a translation. As a consequence, $R=ST$ acts via $[R]:z\lmt \frac{1}{1-z}$, which is a rotation of angle $\frac{\pi}{3}$ about $e^{i\pi/3}$. This leads to the usual construction of a fundamental domain for the action of $\SL(2;\Z)$ (or equivalently $\PSL(2;\Z)$, since the orbits are the same) on $\fh$.

\cF := \big\{ \tau \in \fh \ | \ |\tau| \geq 1 \text{ and } \textstyle \frac{1}{2} \leq \mathrm{Re}\,(\tau) \leq \textstyle\frac{1}{2} \big\}\,.

Then the following properties hold:

* Every $\SL(2;\Z)$-orbit in $\fh$ intersects $\cF$ in at least one point and at most two points.

* If an orbit intersects $\cF$ in two points, then these two points lie on the topological boundary $\partial \cF$ of $\cF$.

* The points of $\cF$ with non-trivial stabiliser in $\SL(2;\Z)$ are the points $i = e^{i\pi/2}$, whose stabiliser is the cyclic group of order $4$ generated by $S$, and the points $j = e^{i2\pi/3}$ and $-j^2 = -\frac{1}{j} = e^{i\pi/3} = j \cdot T$, whose stabilisers are cyclic groups of order $6$, generated respectively by $TRT^{-1} = TS$ and $R$.

Note that the point at infinity of $\CP^1=\C \cup \{\infty\}$ may be added to $\fh \subset \C$ and that the action of $\SL(2;\Z)$ on $\fh$ by biholomorphisms extends to a conformal action on $\hat{\fh}:=\fh\cup\{\infty\}$. The stabiliser of the point at infinity is then the infinite cyclic group generated by $T$. Algebraically, the point at infinity no longer corresponds to an elliptic curve (smooth plane cubic with a marked point) but to a nodal plane cubic with a marked point (given in an affine chart by an equation of the form $y^2 = x^2(x-1)$).

Let $\tau\in\cF$ and let $\Lambda(\tau):=\Z1\oplus \Z\tau$ be the associated lattice in $\C$. Let $G := \Aut(\C/\Lambda(\tau),0)$ be the automorphism group of the elliptic curve $(\C/\Lambda(\tau),0)$. Then the following properties hold:

* If $\tau\neq i, j, -j^2$, then $G\simeq\Z/2\Z$.

* If $\tau=i$, then $G \simeq \Z/4\Z$.

* If $\tau = j$ or $-j^2$, then $G \simeq \Z/6\Z$.

The shape of the fundamental domain $\cF= $ gives us an indication of what the quotient space $\fh/\PSL(2;\Z)$ might look like: a topological space homeomorphic to $\C$, and this identification can be used to put a Riemann surface structure on $\fh/\PSL(2;\Z)$. Thinking about $\fh/\PSL(2;\Z)$ in terms of $\cF$ also suggests that there are three special points, namely $i$, $j$ and $-j^2$, which are the points with non-trivial stabiliser in $\PSL(2;\Z)$, and this makes $\fh/\PSL(2;\Z)$ an orbifold in the sense of Thurston. We will now show that $\fh / \PSL(2;\Z)$ is also the quotient of a Riemann surface by the action of a finite group (Theorem <ref>).

We will recall in Corollary <ref> why $\SL(2;\Z)$ acts properly on $\fh$. By Theorem <ref>, the group $\SL(2;\Z) $ is a finitely generated subgroup of $\GL(2;\C)$. So, by Selberg's Lemma, it contains a torsion-free normal subgroup of finite index, say $\pi \lhd \SL(2;\Z)$. The discrete group $\pi$, being a subgroup of $\SL(2;\Z)$, acts properly on $\fh$. Since $\pi$ is torsion-free, it must therefore act freely on $\fh$ (see Theorems <ref> and  <ref> for further details on that). So the canonical projection $\fh \lra \fh/\pi =: \tilde{M}$ is an analytic covering map between Riemann surfaces, and the topological space $M := \fh/\SL(2;\Z)$ is homeomorphic to $\tilde{M} / \Ga$, where $\Ga:= \SL(2;\Z) / \pi$ is a finite group. A priori, there could exist several pairs $(\tilde{M},\Ga)$ consisting of a Riemann surface $\tilde{M}$ and a finite group $\Ga$ acting on $M$ by automorphisms, such that $\tilde{M}/\Ga \simeq M$. An explicit such presentation is given in Theorem <ref>.

The set $M := \fh/\SL(2;\Z)$ of isomorphism classes of elliptic curves is in bijection with the set $\tilde{M}/\Ga$ where $\tilde{M} = \C\setminus \{0;1\}$ and $\Ga$ is the permutation group $\mathfrak{S}_3 = \left\langle \sigma, \tau\right\rangle $, generated by the permutations $\sigma = (1\ 2)$ and $\tau= (1\ 3)$ and acting to the right on $u \in \C\setminus\{0;1\}$ via $f_\sigma(u) := (1-u)$ and $f_\tau(u) := \textstyle \frac{1}{u}\,$.

As seen after Example <ref>, for all $\tau\in\fh$, the framed elliptic curve $(\C/\Lambda(\tau),0,(1,\tau))$ can be embedded onto the smooth plane cubic of equation

$$y^2 = 4 (x - v_1z) (x-v_2z) (x-v_3z)$$

in $\CP^2$, with pairwise distinct coefficients

v_1 = \wp_{\Lambda(\tau)}\left(\textstyle\frac{1}{2}\right), \  v_2 = \wp_{\Lambda(\tau)}\left(\textstyle\frac{\tau}{2}\right) \ \text{and} \ v_3 = \wp_{\Lambda(\tau)}\left(\textstyle\frac{1+\tau}{2}\right)\in\C\,.
Let us set $u := \textstyle \frac{v_3 - v_2}{v_1 -v_2} \in \C\setminus\{0;1\}$. This corresponds to sending $(v_1,v_2,v_3)$ to $(1,0,u)$ via the homographic transformation $z \lmt \textstyle \frac{z-v_2}{v_1-v_2}$, preserving $\infty\in\CP^1$. So we may also view $u$ as a point of $\CP^1\setminus\{0;1;\infty\}$, with limit cases $u=0,1,\infty$ corresponding respectively to $v_3 = v_2$, $v_3=v_1$ and $v_1=v_2$. The point is that, if $v_1$ and $v_2$ are tranposed by $\sigma = (1\ 2)\in\mathfrak{S}_3$, then $u$ is changed to $f_\sigma(u) = 1-u$, and if $v_1$ and $v_3$ are transposed by $\tau = (1\ 3)\in\mathfrak{S}_3$, then $u$ is changed to $f_\tau(u) = \textstyle \frac{1}{u}$. In fact, the group $\left\langle f_\sigma, f_\tau\right\rangle$ is the group of automorphisms of $\CP^1$ which preserve the subset $\{0;1;\infty\}$. Note that this defines a right action of $\mathfrak{S}_3$ on $\CP^1\setminus\{0;1;\infty\}$ because $f_{\sigma\tau} = f_\tau \circ f_\sigma$. Moreover, if $\tau' \in\fh$, then the corresponding $u'\in \CP^1\setminus\{0;1;\infty\}$ lies in the same $\mathfrak{S}_3$-orbit as $u$ if $\C/\Lambda(\tau) \simeq \C/\Lambda(\tau')$ as elliptic curves. In view of Theorem <ref>, this occurs if and only if $\tau' = \textstyle \frac{a\tau+b}{c\tau+d}$ for some $\textstyle\begin{pmatrix} a & c \\ b & d\end{pmatrix}\in \SL(2;\Z)$. Thus, we have indeed proved that

\begin{equation}\label{quotient_RS}
\fh / \SL(2;\Z) \simeq \big( \C \setminus\{0;1\} \big) / \mathfrak{S}_3 \,.
\end{equation}

The parameter $u\in\C\setminus\{0;1\}$ that appears in the proof of Theorem <ref> coincides with the parameter $u$ in the Legendre family of Example <ref>.

Based on the bijection (<ref>), we can identify this set with the Riemann surface $\C$ in two equivalent ways. Either by defining an $\SL(2;\Z)$-invariant surjection $\hat{j}: \fh\lra\C$ whose fibres are precisely the $\SL(2;\Z)$-orbits in $\fh$, or by defining a $\mathfrak{S}_3$-invariant surjection $\hat{\lambda}: \C\setminus\{0;1\} \lra \C$ whose fibres are precisely the $\mathfrak{S}_3$-orbits in $\C\setminus\{0;1\}$. The first method induces a bijection $j:\fh/\SL(2;\Z)\lra \C$ sending the $\SL(2;\Z)$-orbit of $\tau\in\fh$ to the $j$-invariant of the framed elliptic curve $(\C/\Lambda(\tau),0,(1,\tau))$. The latter is constructed via the theory of modular forms and defined by converting Equation (<ref>) into the equation

\wp^{\prime 2}(z)=4 \wp^3(z)-g_2(\tau) \wp(z)-g_3(\tau)

where $g_2(\tau) = 60 G_4(\tau)$, $g_3(\tau) = 140 G_6(\tau)$ and $G_k(\tau)$ is the Eisenstein series
$$ \sum_{(m,n)\in\Z^2, (m,n)\neq (0,0)} \frac{1}{(m\tau+n)^k}\,,$$
then setting $j(\tau) = \textstyle\frac{1728\, g_2(\tau)^3}{g_2(\tau)^3 - 27 g_3(\tau)^2}$. For the former approach and the definition of a bijection $\lambda : \C\setminus\{0;1\} / \mathfrak{S}_3 \lra \C$, recall that we want a surjective map $\hat{\lambda}: \C\setminus\{0;1\} \lra \C$ with generic fibres having six elements, and that is invariant under the transformations $f_\sigma: u\lmt (1-u)$ and $f_\tau: u \lmt \textstyle \frac{1}{u}$. This leads to $\hat{\lambda}(u) := \textstyle \frac{(1 + u(u-1))^3}{(u(u-1))^2}$, whose numerator is a polynomial of degree $6$ which is non-zero when $u=0$ or $u=1$. It can be checked that, indeed, $\hat{\lambda} \circ f_\sigma = \hat{\lambda}$ and $\hat{\lambda} \circ f_\tau = \hat{\lambda}$. Finally, note that if we reduce the cubic equation $y^2 = x(x-1)(x-u)$ to an equation of the form $y^2 = 4x^3 -g_2x - g_3$, we find that
$g_2 = \textstyle \frac{\sqrt[3]{4}}{3}(1 + u (u-1))$ and $g_3 = \textstyle \frac{1}{27}(u+1)(2 u^2-5 u+2)$, so the $j$-invariant of the elliptic curve defined by the Legendre equation $y^2 = x(x-1)(x-u)$ is equal to $256 \times \hat{\lambda}(u)$.

§ STACKS

§.§ Prestacks

Prestacks formalise the notion of families of geometric objects by building upon the intuition that a family is a special kind of map, one that you should be able to pull back. For instance, if a group $G$ acts on a space $X$, there should be:

* For all space $S$, a notion of family of $G$-orbits in $X$, parameterised by $S$, and a notion of isomorphism of such families.

* For all morphism $f:T\lra S$ and all family $P$ parameterised by $S$, a family $f^*P$ parameterised by $T$, such that there are natural isomorphisms $\id_S^*P\simeq P$ and $(f\circ g)^*P \simeq g^*(f^*P)$ for all morphism $g:U\lra T$.

If $G$ is a Lie group acting by automorphisms on a manifold $X$, a family parameterised by $S$ of $G$-orbits in $X$ is defined to be a pair $(P,u)$ where $P$ is a principal $G$-bundle on $S$, and $u:P\lra X$ is a $G$-equivariant map. In particular, the image of $u$ is a union of $G$-orbits in $X$, and the intersection of the fibre of $u$ above a point $x\in X$ with a $G$-orbit in $P$, if non-empty, is in bijection with the stabiliser of $x$ in $G$. Moreover, if $f:T\lra S$ is a morphism, then $f^*P$ is indeed a principal $G$-bundle on $T$, equipped with a $G$-equivariant map to $X$:

\begin{equation*}
\begin{tikzcd}
f^*P \ar[r] \ar[d] & P \ar[r, "u"] \ar[d] & X\\
T \ar[r, "f"] & S &
\end{tikzcd}
\end{equation*}

An isomorphism between two families $(P,u)$ and $(Q,v)$ parameterised by the same space $S$, is an isomorphism $\lambda:P\overset{\simeq}{\lra} Q$ of principal $G$-bundles over $S$, such that $v\circ\lambda = u$. Note that the definitions above make sense when $X=\bullet$ (a single point), with $G$ acting trivially on it.

We are now ready to give the formal definition of a prestack. Since we are mainly interested in the moduli stack of elliptic curves, our spaces $S$ will be complex analytic manifolds. We shall denote by $\An$ the category of such spaces.

A prestack on $\An$ is a $2$-functor $\M: \An^{\mathrm{op}} \lra \text{Groupoids}$, meaning an assignment,

* for all complex analytic manifold $S\in\An$, of a groupoid $\M(S)$;

* for all morphism $f:T\lra S$ in $\An$, of a pullback functor $f^{\,*}_\M:\M(S)\lra \M(T)$, often denoted simply by $f^*$;

* for all pair of composable morphisms $g:U\lra T$ and $f:T\lra S$ in $\An$, of a commutative diagram

\begin{equation}\label{identification_of_composed_pullback}
\begin{tikzcd}
\M(S) \ar[r, "f^*"] \ar[d, "(f\circ g)^*"] & \M(T) \ar[d, "g^*"] \\
\M(U) \ar[r, "\Phi^{\,\M}_{f,g}"] & \M(U)
\end{tikzcd}
\end{equation}

such that:

* $\id_S^* = \id_{\M(S)}$, and

* for all morphism $h:V\lra U$ in $\An$, the following diagram commutes:

\begin{equation}\label{compatibility_of_pullbacks}
\begin{tikzcd}
& \M(S) \ar[rd, "f^*"] \ar[ld, "(f\circ g)^*"] \ar[ddd, dashed, "(f\circ g\circ h)^*"]&\\
\M(U) \ar[d, "h^*"] & & \M(T) \ar[d, "(g\circ h)^*"] \\
\M(V) \ar[rd, "\Phi^{\,\M}_{f\circ g,h}"] & & \M(V) \ar[ld, "\Phi^{\,\M}_{f,g\circ h}"] \\
& \M(V) &
\end{tikzcd}
\end{equation}

In other words, the dashed arrow $(f\circ g\circ h)^*:\M(S) \lra \M(V)$ is well-defined.

In practice, the identification $(f\circ g)^*\simeq_{\Phi^{\,\M}_{f,g}} g^*\circ f^*$ in Diagram (<ref>) is often canonical: this was for instance the case in Example <ref> where, for all principal bundle $P\lra S$, it follows from the construction of the pullback bundle that there is indeed a canonical isomorphism $(f\circ g)^*P = g^*(f^*P)$. Likewise $h^*((f\circ g)^*P) = (g\circ h)^* ( f^* P)$ , which means that Condition (<ref>) is automatically satisfied in this case. So Example <ref> indeed defines a prestack, usually denoted by $[X/G]$ and called the quotient stack of $X$ by $G$. If $\M$ is a prestack and $S$ is a manifold, the groupoid $\M(S)$ is called the groupoid of $S$-points of $\M$. For instance, the groupoid of $S$-points of $[X/G]$ is
$$[X/G](S) := \left\{
\begin{tikzcd}
P \ar[r, "u"'] \ar[d] & X\\
S &
\end{tikzcd}
\begin{array}{l}
\text{with}\ u:P\to X\ \text{a}\ G\text{-equivariant map}\\
\text{and}\ P\to S\ \text{a\ principal}\ G\text{-bundle}
\end{array}
\right\}.$$

When $X=\bullet$ is a point, the prestack $[\bullet/G]$ is usually denoted by $BG$, and called the classifying stackstack!classifying stack of $G$. The case when $G=\{1\}$ is also interesting, for it enables us to see manifolds as prestacks (Example <ref>). Finally, as we shall see in Section <ref>, families of elliptic curves parameterised by a complex analytic manifold $S\in\An$ indeed define a prestack.

A manifold $X$ defines a prestack, denoted by $\underline{X}$, by setting $\underline{X}(S):=\mathrm{Mor}(S,X)=\{u:S\lra X\}$, with morphisms between $u:S\lra X$ and $u':S \lra X$ given by

$$ \Mor_{ \underline{ X }( S ) }( u, u' ) :=
\left\{
\begin{array}{cl}
\{\id_S\} & \text{if}\ u=u' \, , \\
\emptyset & \text{if}\ u \neq u' \, .
\end{array}
\right.
Note that this is the standard way in which groupoids generalise sets: the groupoid associated to a set is the category whose objects are the elements of the set and the only morphisms are the identity morphisms. The pullback functor corresponding to $f:T\lra S$ is given by $f^*u=u\circ f$, which is indeed a functor given the way morphisms are defined in $ \underline{X}( S ) $ and $ \underline{X}( T ) $. As we shall see in Lemma <ref>, this defines a fully faithful functor $X\lmt \underline{X}$ from manifolds to stacks. A prestack $\X$ for which there exists a manifold $X$ such that $ \X \simeq \underline{X} $ is called a representable stackstack!representable stackrepresentable!stack. By Lemma <ref>, a manifold $X$ satisfying $\underline{X}\simeq \X$ is defined up to canonical isomorphism.

To conclude this subsection, we give the definition of a morphism of prestacks.

Let $\M,\N:\An^{\mathrm{op}}\lra\text{Groupoids}$ be prestacks on $\An$. A morphism of prestacks $F:\M\lra \N$ is a a natural transformation between the $2$-functors $\M$ and $\N$, meaning an assignment for all space $S\in\An$, of a functor $F_S:\M(S)\lra \N(S)$ such that, for all morphism $f:T\lra S$ in $\An$, the following diagram is $2$-commutative (i.e. commutative up to a natural transformation in the category $ \N( T ) $)

\begin{equation}\label{2_comm_diag}
\begin{tikzcd}
\M(S) \ar[r, "F_S"] \ar[d, "f^{\,*}_\M"] & \N(S) \ar[d, "f^{\,*}_\N"]\\
\M(T) \ar[r, "F_T"] & \N(T)
\end{tikzcd}
\end{equation}

and moreover, for all $g:U\lra T$ in $\An$, we have $F_U(\Phi^{\,\M}_{f,g}) = \Phi^{\,\N}_{F_U(f),F_U(g)}$ as functors from $\N(U)$ to $\N(U)$. We shall denote by $\mathcal{PS}t$ the $2$-category of prestacks.

If $X$ is a manifold, morphisms of prestacks $F\in\Mor_{\mathcal{PS}t}(\underline{X},\M)$ correspond to objects $P_X\in\M(X)$. Indeed, the morphism $F$ is entirely determined by the object $P_X:=F_X(\id_X)\in \M(X)$ and the fact that, as $\id_X \circ u = u$, the morphism $F$ must satisfy, for all $S\in\An$,

\underline{X}(S)=\Mor_{\An}(S,X) & \lra & \M(S)\\
u & \lmt & u^*_\M P_X
\end{array}\, .
This sets up an equivalence of categories $\Mor_{\mathcal{PS}t}(\underline{X},\M) \simeq \M(X)$, obtained by sending $F$ to $P_X:=F_X(\id_X)$. For instance, if $\M=BG:=[\bullet/G]$, the groupoid $BG(S)$ is the category of principal
$G$-bundles on $S$ and, given a principal bundle $P\lra X$, there is a natural functor

$$F_S: \begin{array}{rcl}
\Mor_{\An}(S,X) & \lra & BG(S)\\
u & \lmt & u^* P
\end{array}
and the $2$-commutativity of Diagram (<ref>) holds because of the canonical identification $(u\circ f)^*P \simeq f^*(u^*P)$ for all $f:T\lra S$.

Unwrapping the details of Example <ref>, we get the following stacky version of the Yoneda lemma.

Let $X,Y$ be objects in $\An$ and let $\underline{X}, \underline{Y}$ be the corresponding prestacks. Then
$$\Mor_{\mathcal{PS}t}(\underline{X},\underline{Y}) \simeq \Mor_{\An}(X,Y).$$
In particular, $\underline{X}\simeq \underline{Y}$ in $\mathcal{PS}t$ if and only if $X\simeq Y$ in $\An$.

§.§ Stacks

There is an obvious analogy between Definition <ref> and that of a presheaf on $\An$. Based on that analogy, we can define stacks as prestacks that satisfy a descent condition with respect to open coverings of objects $S\in\An$. The only subtlety is that we have to glue not only morphisms but also objects. In what follows, given a prestack $\M$ on $\An$, a morphism $f:T\lra S$ in $\An$, and a morphism $\phi:A\lra B$ in $\M(S)$, we denote by $A|_T$ and $B|_T$ the objects $f^*A$ and $f^*B$ of $\M(T)$, and by $\phi|_T:A|_T\lra B|_T$ the morphism $f_\M^*\phi$ in $\M(T)$.

A stack on $\An$ is a prestack $\M:\An^{\mathrm{op}}\lra\text{Groupoids}$ such that, for all $S\in\An$ and all open covering $(S_i)_{i\in I}$ of $S$, the following two conditions are satisfied:

* (Gluing of morphisms). For all pair of objects $A,B\in\M(S)$ and all family of morphisms
$$\phi_i:A|_{S_i}\lra B|_{S_i}\ \text{in}\ \M(S_i)$$
such that, for all $ i, j$ in $I$, $ \phi|_{ S_i \cap S_j } = \phi_j|_{ S_i \cap S_j } $, there exists a unique morphism $\phi:A\lra B$ in $\M(S)$ such that, for all $i\in I$, $\phi|_{S_i}=\phi_i$.

* (Gluing of objects). For all family of objects $(A_i)_{i\in I} \in \prod_{i\in I}\M(S_i)$ equipped with isomorphisms
$$\phi_{ij}: A_j|_{S_i\cap S_j} \overset{\simeq}{\lra} A_i|_{S_i\cap S_j}\ \text{in}\ \M(S_i\cap S_j)$$
satisfying the cocycle conditions $\phi_{ii} = \id_{A_i}$ and $\phi_{ij}\circ\phi_{jk} = \phi_{ik}$ in $\M(S_i\cap S_j\cap S_k)$ for all triple $(i,j,k)$, there exists an object $A\in\M(S)$ together with isomorphisms $\phi_i:A|_{S_i}\lra A_i$ in $\M(S_i)$, such that, for all $(i,j)$, one has $ \phi_i \circ\phi_j^{-1} = \phi_{ij}$ in $\M(S_i\cap S_j)$.

A morphism of stacks $F:\M\lra\N$ is a morphism of prestacks from $\M$ to $\N$. We shall denote by $\mathcal{S}t$ the $2$-category of stacks.

It is perhaps not easy to note at first, but an important feature of the definition of stacks is that objects are required to glue but without any requirement of uniqueness: in Point (2) of Definition <ref>, the object $ A $ constructed from the $ A_i $ is “unique up to isomorphism", by a direct application of Condition (1), but the isomorphisms $ \phi_i: A|_{ S_i } \lra A_i $ are not unique, which is the familiar condition from the theory of fibre bundles and is in fact what allows objects with non-trivial automorphisms to define stacks. We will for instance study the stack of elliptic curves from that point of view in Section <ref>.

The prestacks $[X/G]$, $BG$ and $\underline{X}$ introduced in Examples <ref> and <ref> are stacks on $\An$.

The prestacks $BG$ and $\underline{X}$ are special cases of the prestack $[X/G]$. We leave the proof that the prestack $[X/G]$ is a stack as an exercise (it follows from the fact that principal $G$-bundles on $S\in\An$, and morphisms between them, can be constructed by gluing).

In line with the intuition that stacks are designed to formalise the idea of families as mathematical objects that can be pulled back (prestack) and glued (stacks), the $2$-category of stacks admits fibre products.

fibre product of stacks
Let $F:\M\lra \X$ and $G:\N\lra \X$ be morphisms of stacks on $\An$. Let us denote by $\M \times_\X \N$ the $2$-functor sending a space $S\in\An$ to the groupoid  whose objects are

\begin{equation}\label{fibre_product_def}
\Mor_{\mathcal{S}t}(\underline{S},\M\times_\X\N) :=
\left\{
(\alpha,\beta,\phi)\quad \Bigg|\quad \substack{\displaystyle{\alpha:\underline{S}\lra \M,}\\ \displaystyle{\beta: \underline{S}\lra \N,}\\ \displaystyle{\phi:\X\overset{\simeq}{\lra} \X}}\ \text{such that}
\begin{tikzcd}
& \X \ar[dd, "\phi", "\simeq"'] \\
\underline{S} \ar[ur, "F\circ \alpha"] \ar[dr, "G\circ \beta"'] & \\
& \X
\end{tikzcd}\
\text{commutes.}
\right\}
\end{equation}
and whose morphisms between the objects $(\alpha,\beta,\phi)$ and $(\alpha',\beta',\phi')$ are given by triples $(\lambda,\mu,\psi)$ of morphisms $\lambda:\M\lra\M$, $\mu:\N\lra\N$ and $\psi:\X\overset{\simeq}{\lra}\X$ such that $F\circ(\lambda\circ\alpha) = \psi \circ (F\circ\alpha)$ and $G\circ(\mu\circ\beta) = \psi \circ (G\circ\beta)$ as morphisms from $\underline{S}$ to $\X$, and $\phi'\circ\psi=\psi\circ\phi$ as morphisms from $\X$ to $\X$ (i.e. $(\lambda,\mu,\psi)$ induces an isomorphism between diagrams of the form appearing in (<ref>)).

Definition <ref> is not easy to digest, because it involves a lot of hidden compatibility conditions. It is instructive to compare it to Thurston's definition of the fibre product of two orbifolds, which appears in the course of the proof of Proposition 13.2.4 in <cit.>. If we think of the groupoid $\Mor_{\mathcal{PS}t}(\underline{S},\M\times_\X\N)$ as families parameterised by $S$, we get the following more concrete description:

\begin{equation}\label{fibre_product_alternate_def}
(\M \times_\X \N)(S) := \big\{(A,B,\phi_S)\ |\ A\in\M(S), g\in\N(S)\ \text{and}\ \phi_S:F_S(A)\overset{\simeq}{\lra} G_S(B)\ \text{in}\ \X(S)\big\}
\end{equation}
with morphisms $(A,B,\phi_S)\lra (A',B',\phi'_S)$ being pairs $(u_S,v_S)$ of morphisms $u_S:A\lra A'$ in $\M(S)$ and $v_S:B\lra B'$ in $\N(S)$ making the following diagram commute in $\X(S)$:

\begin{equation*}
\begin{tikzcd}
F_S(A) \ar[r, "\phi_S", "\simeq"'] \ar[d, "F_S(u_s)"] & G_S(B) \ar[d, "G _S(v_S)"] \\
F_S(A') \ar[r, "\phi'_S", "\simeq"'] & G_S(B')
\end{tikzcd}
\end{equation*}
Let us show how we can use (<ref>) as a definition of the fibre product $\M\times_\chi\N$. We simply observe that, because of the fact that, for all morphism $f:T\lra S$ in $\An$, we have $F_T(A|_T) = F_S(A)|_T$ and $G_T(B_T)=G_S(B)|_T$ in $\X(T)$, we must add the compatibility condition that $\phi_T=\phi_S|_T$ as functors from $\X(T)$ to $\X(T)$, to ensure that we have a well-defined pullback morphism

f^{\ *}_{\M\times_\X\N} :
\begin{array}{rcl}
(\M\times_\X\N)(S) & \lra & (\M\times_\X\N)(T)\\
(A,B,\phi_S) & \lmt & (A|_T, B|_T,\phi_T)
\end{array}\,.
This compatibility condition, which needs to be added here, is encapsulated in Definition <ref> as the fact that the arrow $\phi:\X\lra\X$ appearing in Diagram (<ref>) is a morphism of prestacks.

Let us compute the fibre product of a morphism $\underline{M}\lra BG$ with the morphism $\underline{\bullet}\lra BG$, where $M\in\An$ and $\bullet$ is a single point. By Example <ref>, the first morphism corresponds to a principal $G$-bundle $P\lra M$, while the second morphism corresponds to the $G$-bundle $G\lra\bullet$. Then, by definition of the stacks $\underline{M}$, $\underline{\bullet}$ and $\underline{M}\times_{BG}\bullet$, and using the fact that a principal bundle is trivial if and only if it admits a global section, we have:

\begin{eqnarray*}
(\underline{M}\times_{BG}\underline{\bullet})(S) & \simeq & \left\{ (u,\phi)\ |\ u:S\lra M\ \text{and}\ \phi\ \text{is\ an\ isomorphism}\ u^*P\lra S\times G\right\}\\
& \simeq & \{ (u,\sigma)\ |\ u:S\lra M\ \text{and}\ \sigma\ \text{is\ a\ section\ of}\ u^*P\}.
\end{eqnarray*}
But, by the universal property of $u^*P$, giving a section $\sigma:S\lra u^*P$ is equivalent to giving a morphism $f:S\lra P$.

\begin{tikzcd}
S \ar[ddr, bend right=20, "\id_S"'] \ar[rrd, bend left=20, "f"] \ar[rd, "\exists!\,\sigma", dashed]& & \\
& u^*P \ar[r] \ar[d] & P \ar[d] \\
& S \ar[r] & M
\end{tikzcd}
So $(\underline{M}\times_{BG}\underline{\bullet})(S) \simeq \Mor(S,P)$, which proves that the fibre product $\underline{M}\times_{BG}\bullet$ is isomorphic to the stack $\underline{P}$, where $P\lra M$ is the principal $G$-bundle defining the given morphism $\underline{M}\lra BG$. In particular, the stack $\underline{M}\times_{BG}\bullet$ is representable by the manifold $P$.

The $2$-functor $\M\times_\X\N$ defined in (<ref>) is a prestack and it comes equipped with two morphisms $p_\M:\M\times_\X\N\lra \M$ and $p_N:\M\times_\X\N\lra \N$ making the following diagram a $2$-cartesian square:

\begin{equation*}
\begin{tikzcd}
\M\times_\X\N \ar[r, "p_\N"] \ar[d, "p_\M"] & \N \ar[d, "G"]\\
\M \ar[r, "F"] & \X
\end{tikzcd}
\end{equation*}

We deduce from the discussion following Definition <ref> that $\M\times_\chi\N$ is a prestack. By (<ref>), the prestack $\M\times_\chi\N$ satisfies the universal property of a $2$-fibre product for all prestacks of the form $\underline{S}$ where $S\in\An$. Therefore, it satisfies it for all prestacks on $\An$.

We omit the proof that the prestack $\M \times_\X \N$ is a stack. Note that if $\underline{M}, \underline{N}$, and $\underline{X}$ are representable stacks, the stack $\underline{M}\times_{\underline{X}} \underline{N}$ is not representable in general, because $M\lra X$ and $N\lra X$ do not admit a fibre product in $\An$ in general.

§.§ Analytic stacks
An analytic stack is a stack $\X$ on $\An$ that admits a presentation (or atlas) $p:\underline{X}\lra \X$, where $\underline{X}$ is the stack associated to a space $X\in\An$ (henceforth, we shall often write simply $X$ for the stack $\underline{X}$ associated to the space $X\in\An$). Before we can define more precisely what a presentation is, we need to introduce the following notion.

A morphism of stacks $F:\M\lra\X$ is called representable if, for all morphism $S\lra \M$ with $S\in\An$, there exists a complex analytic manifold $M_S\in\An$ such that $S\times_\X \M \simeq M_S$.

In other words, for all $S\lra \X$, the morphism of stacks $F:\M\lra\X$ base-changes to a morphism of complex analytic manifolds $f_S:M_S\lra S$, as per the following $2$-Cartesian square:

\begin{equation}\label{base_change_rep_morphism}
\begin{tikzcd}
S\times_\X\M \simeq M_S\ar[d, "f_S"] \ar[r] & \M \ar[d, "F"] \\
S \ar[r] & 	\X
\end{tikzcd}
\end{equation}

The computation of $M\times_{BG}\bullet$ in Example <ref> shows that the canonical morphism $\bullet\lra BG$ is representable. Since the pullback of the trivial bundle $(X\times G)\lra X$ to an arbitrary manifold $S$ mapping to $X$ is trivial, the same proof works for the canonical morphism $X\lra [X/G]$, corresponding to the trivial bundle $X\times G$ (with $G$-equivariant map from $X\times G$ to $X$ given by the action map $(X\times G)\lra X$, where $G$ acts on $X$ to the right). Note that the forgetful morphism $BG\lra\bullet$ (sending a principal $G$-bundle $P\lra S$ to the constant map $S\lra\bullet$) is not a representable morphism as soon as $\bullet\times_\bullet BG \simeq BG$ is not a representable stack.

Working with representable morphisms enables one to carry over certain local properties of morphisms from $\An$ to $\mathcal{S}t$. Most prominently for us, the following one.

Let $\M$ and $\X$ be stacks over $\An$. A morphism of stacks $F:\M\lra\X$ is called a submersion if it is representable and, for all morphism of stacks $S\lra \X$ with $S\in\An$, the induced morphism $f_S:M_S\lra S$ appearing in Diagram (<ref>) is a submersion in $\An$.

Intuitively, this is a meaningful definition because, for manifolds, a morphism $f:Y\lra X$ is a submersion in $\An$ if and only if for all open coverings $X = \cup_{i\in I} X_i$ and $Y=\cup_{j\in J} Y_j$, the induced morphisms $f_{ij}: Y_j\cap f^{-1}(X_i) \lra X_i$ are submersions. Next, we define surjective morphisms of stacks (which bears an analogy to the notion of surjective morphism of sheaves).

Let $\M$ and $\X$ be stacks over $\An$. A morphism $F:\M\lra\X$ is said to be surjective if, for all $S\in\An$ and all $P\in \X(S)$, there is an open covering $(S_i)_{i\in I}$ of $S$ such that $P|_{S_i}$ is in the essential image of the functor $F_{S_i}:\M(S_i)\lra \X(S_i)$, i.e. there exists a family $(P_i)_{i\in I}$ of objects $P_i\in \M(S_i)$ and, for all $i\in I$, an isomorphism $F_{S_i}(P_i) \simeq P|_{S_i}$ in $\X(S_i)$.

We can now define a presentation (or atlas of an analytic stack)atlas of a stack $\X$ on $\An$ as a surjective representable morphism $p:X\lra \X$ which is a submersion and whose domain is a manifold $X\in\An$. This leads to the following definition of an analytic stack.

stack!analytic stack
An analytic stack is a stack $\X$ on $\An$ equipped with an analytic manifold $X\in\An$ and a morphism $p:X\lra\X$ such that:

* $p$ is representable (Definition <ref>).

* $p$ is surjective (Definition <ref>).

* $p$ is a submersion (Definition <ref>).

A morphism between analytic stacks $\M\lra \X$ is a morphism of stacks from $\M$ to $\X$.

The same definition works for a differentiable stack $\X$; it is a stack on the category of differentiable manifolds which is equipped with a representable surjective morphism $p:X\lra \X$ such that $X$ is a differentiable manifold and $p$ is a submersion.

It is straightforward to check that, if $X\in\An$, the identity morphism $\id_X$ gives a presentation $X\lra \underline{X}$. So $\underline{X}$ is an analytic stack in a canonical way. In fact, we could also take, as a presentation for $\underline{X}$, the analytic manifold $\sqcup_{i\in I}U_i$, where $(U_i)_{i\in I}$ is an analytic atlas of $X$ in the traditional sense. Let us now prove that if $G$ is a complex Lie group acting on an analytic manifold $X$, then the quotient stack $[X/G]$ admits a canonical structure of analytic stack, given by the morphism $X\lra[X/G]$ defined by the trivial principal $G$-bundle on $X$. We saw in Example <ref> that this morphism is representable. It is surjective because principal $G$-bundles on a manifold $S$ are locally trivial. And it is a submersion because the projection map of a principal bundle $P\lra S$ is a submersion. In particular, the classifying stack $BG$ of a complex Lie group $G$ admits a canonical structure of analytic stack.

Note that a fibre product of analytic stacks is a stack over $\An$ but not necessarily an analytic stack.

§ ORBIFOLDS

We will eventually define an analytic orbifold as an analytic stack admitting an open covering by quotient subststacks of the form $[U/\Gamma]$, where $U$ is an analytic manifold and $\Gamma$ is a finite group acting on $U$ by automorphisms. Before we can do that, we need to define open substacks.

Let $\X$ be a stack on $\An$. A substack of $\X$ is a pair $(\Y,F)$ where $\Y$ is a stack on $\An$ and $F:\Y\lra \X$ is a morphism of stacks. If $\X$ is an analytic stack and the substack $\Y$ is also analytic, we will call $\Y$ an analytic substack of $\X$.

Let $G$ be a complex Lie group acting to the right on an analytic manifold $X$ and let $Y\subset X$ be an open submanifold of $X$ that is invariant under the action of a subgroup $H\subset G$. The $H$-equivariant map $Y\hookrightarrow X$ induces a canonical morphism of stacks $F:[Y/H]\lra [X/G]$. Explicitly, the morphism $F$ is defined, for all $S\in\An$, by the functor

\begin{equation}\label{ext_struct_gp}
\left(
\begin{tikzcd} P \ar[r] \ar[d] & Y\\ S & \end{tikzcd}
\right)
\lmt
\left(
\begin{tikzcd} (P\times_H G) \ar[r] \ar[d] & Y\cdot G \underset{\text{\tiny{open}}}{\subset} X\\ S & \end{tikzcd}
\right)
\end{equation}
sending a principal $H$-bundle $P$ on $S$ equipped with an $H$-equivariant map to $Y$ to the extension of structure group $P\times_H G$, equipped with the induced $G$-equivariant map to $Y\cdot G\subset X$ . This morphism $F$ makes $[Y/H]$ a substack of $[X/G]$. Note that $[X/G](S)$ is in general strictly larger than $[Y/H](S)$, since it is possible that not all principal $G$-bundles on $S$ admit a reduction of structure group on $H$. Also, the functor $F_S$ is in general neither full nor faithful: given two principal $G$-bundles on $S$ admitting a reduction of structure group to $H$, a morphism between them does not necessarily come from a morphism between the two reductions and, even if it does, the morphism it comes from is not necessarily unique (for instance, a given principal $G$-bundle may admit non-isomorphic reductions of structure group to $H$). Note that if we study the analogous situation with $Y$ closed in $X$, we need to be careful with the manifold structure on $Y\cdot G$ in order to get a morphism $Y\cdot G\lra X$ in $\An$ (which may or may not be a closed immersion).

It is easy to make sense of the notion of open substack for representable morphisms (Definition <ref>). And for analytic stacks, it suffices to check the condition with respect to an atlas (Proposition <ref>).

Let $\X$ be a stack on $\An$. A substack $F:\Y\lra\X$ is called an open substacksubstack!open substack of $\X$ if:

* $F$ is representable (Definition <ref>).

* For all morphism of stacks $M\lra \X$ with $M\in\An$, the morphism $f_M: \Y\times_\X M \lra M$ is an open morphism in $\An$.

Note that Condition 2, which says that $f_M: \Y\times_\X M \lra M$ is open in $\An$, makes sense because the stack $\Y\times_\X M$ is representable by a manifold (in this case because $F$ is assumed to be a representable morphism; see Remark <ref> for further comments).

Let us retake Example <ref>. We will show that, if $Y\subset X$ is open and $G/H$ is a discrete topological space, then $[Y/H]$ is an open substack of $[X/G]$. This will apply in particular when $G$ is a discrete group (see also Example <ref>). Firstly, we have to show that the morphism (<ref>) is representable and secondly, that it is open (this is where we shall use that $G/H$ is discrete). For the first part, we compute the fibre product $[Y/H]\times_{[X/G]} M$ for all morphism of stacks $M\lra [X/G]$ with $M\in\An$. Such a morphism is given by a principal $G$-bundle $P$ over $M$, equipped with a $G$-equivariant morphism $P\lra X$ in $\An$. Likewise, an arbitrary morphism $S\lra[Y/H]$ with $S\in\An$ is given by a principal $H$-bundle $Q$ over $S$, equipped with an $H$-equivariant morphism $Q\lra Y$ in $\An$. So, given a morphism $u:S\lra M$ in $\An$, a morphism of stacks $S\lra [Y/H]\times_{[X/G]} M$ is given by an isomorphism of principal $G$-bundles between $u^*P$ and $Q\times_H G$ over $S$, i.e. a reduction of structure group from $G$ to $H$ for the principal $G$-bundle $u^*P\lra S$. It is well-known that reductions of structure group are in bijection with sections of the associated fibre bundle $u^*P\times_G (G/H) \simeq u^*(P\times_G (G/H))$ over $S$. Then, as in Example <ref>, such sections are in bijection with morphisms of analytic manifolds from $S$ to $P \times_G (G/H)$, which proves that the canonical morphism $F:[Y/H]\lra[X/G]$, defined in (<ref>) by extension of structure group and inclusion of $Y\cdot G$ in $X$, is representable (in particular, when $M=X$ and $P=X\times G$, then $P\times_G(G/H) = X\times (G/H)$). Since we now have a $2$-Cartesian diagram

\begin{tikzcd}
P \times_G (G/H)\simeq \left[Y/H\right]_M \ar[r] \ar[d] & \left[Y/H\right] \ar[d] \\
M \ar[r, "P"'] & \left[X/G\right]
\end{tikzcd}
we see that the morphism of stacks $[Y/H]\lra[X/G]$ pulls back to the fibre bundle $P\times_G (G/H) \lra M$ in $\An$, which is indeed an open map since the fibre, which is $G/H$, is discrete.

Note that, if we take $H=G$ in Example <ref>, then the canonical morphism $[Y/G]\lra [X/G]$ is an open embedding in the sense that, for all $M\in\An$, the morphism $[Y/G]_M \lra M$ is an open embedding in $\An$ (this is obvious because $[Y/G]_M \simeq M$ when $H=G$).

Let $\X$ be a stack on $\An$ and assume that $\X$ is an analytic stack with presentation $p:X\lra\X$. A substack $F:\Y\lra\X$ is an open substack of $\X$ if:

* $F$ is representable.

* The morphism $f_X: \Y\times_\X X \lra X$ is an open morphism in $\An$.

If $(\Y,F)$ is an open substack of the analytic stack $\X$, Conditions (1) and (2) of Proposition <ref> are indeed met. To prove the converse, it suffices to show that, if $f_X:Y_X\lra X$ is open, then $f_M:Y_M\lra M$ is open for all $M\in\An$, where, for all $S\in\An$, we use the notation $Y_S\in\An$ for the representable stack $\Y\times_\X S$. To prove what we need to, note that, given a morphism of stacks $M\lra \X$, we have a commutative diagram

\begin{tikzcd}
Y_X\times_X X_M \ar[d, "\text{open}"] \ar[r,"\simeq"] & \Y\times_\X X_M \ar[r] \ar[d, "\text{open}"] & \Y\times_\X M \ar[d] \ar[r] & \Y \ar[d]\\
X_M \ar[r, equal] & X_M \ar[r, "\overset{\text{\tiny{surjective}}}{\text{\tiny{submersion}}}"] & M \ar[r] & \X
\end{tikzcd}
where the first vertical arrow is open because $f_X:Y_X\lra X$ is open by the assumption made on $F:\Y\lra \X$ and since the property of being open is invariant by base change. The commutativity of the left square then implies that the second vertical arrow is open. Since $F:\Y\lra \X$ is a representable morphism of stacks, the stack $\Y\times_\X M$ is representable by a manifold, and since the middle square is a pullback diagram, the morphism $Y\times_\X X_M \lra Y\times_\X M$ is also a surjective submersion in $\An$. In particular, it is open and surjective. So the fact that $Y\times_\X X_M\lra X_M$ is open implies that $f:\Y\times_\X M \lra M$ is indeed open.

We leave it as an exercise to show that, if $F:\Y\lra\X$ is an open substack of an analytic stack $\X$, then $\Y$ is an analytic stack.

Some comments are in order. For the sake of simplicity, we have omitted certain aspects of the definition of an analytic stack or of an open substack. For instance, in Definition <ref>, it is often required that, for a stack $\X$ over $\An$ to be an analytic stack, every morphism $M\lra\X$ with $M$ in $\An$ should be representable. So Condition (1) of Definition <ref> becomes unnecessary. But of course it then becomes more difficult to show that a given stack over $\An$ is an analytic stack. In practice, one uses the fact that the following two conditions are equivalent (we refer to <cit.> for a proof of this):

* Every morphism $M\lra\X$ with $M$ in $\An$ is representable (meaning that, for all $N\lra\X$ with $N\in\An$, the stack $M\times_\X N$ is representable by a manifold).

* The diagonal morphism $\Delta:\X\lra\X\times\X$ is representable.

So the more official version of Definition <ref> is that a stack $\X$ on $\An$ is called an analytic stackstack!analytic stack if the diagonal morphism $\Delta:\X\lra\X\times\X$ is representable and there exists a manifold $X\in\An$ and a (necessarily representable) surjective submersion $p: X \lra \X$. The stack $\X$ is called separatedstack!separated stack if the representable morphism $\Delta$ is closed.

In the same way that we defined what it means to be an open substack of a stack $\X$, we can define what it means to be a cover of such a stack. Comments similar to Proposition <ref> and Remark <ref> apply when $\X$ is an analytic stack, but we omit them here and only give the elementary definition.

Let $\X$ be a stack on $\An$. A morphism of stacks $F:\Y\lra\X$ is called a cover or covering stackstack!covering stack of $\X$ if:

* $F$ is representable.

* For all morphism of stacks $M\lra \X$ with $M\in\An$, the morphism $f_M: \Y\times_\X M \lra M$ is a cover in $\An$.
Such a cover is called a finite cover if, for all $M$, the covering map $f_M$ is a finite cover in $\An$.

If $G$ is a discrete group acting by automorphisms on the space $X\in\An$, then the canonical morphism $X\lra [X/G]$ (defined by the trivial cover $(X\times G)\lra X$) is a covering stack of $[X/G]$. Indeed, if $M\in\An$ and $P\lra M$ is the principal $G$-bundle associated to a morphism $M\lra [X/G]$, then $X\times_{[X/G]} M$ is isomorphic to $P$ (see Examples <ref> and <ref>), and this is indeed a cover of $M$ because $G$ is discrete. A subgroup $H< G$ then defines an intermediate cover $X\lra [X/H] \lra [X/G]$. And if $Y$ is a $G$-equivariant cover of $X$, then $[Y/G]\lra [X/G]$ is a cover in the stacky sense. Mixing these two examples together, we get the following commutative diagrams, all of whose arrows are covers (the first vertical arrow being a $G$-equivariant cover in $\An$).

\begin{tikzcd}
Y \ar[r] \ar[d, "H< G\ \Circlearrowright"'] & \left[Y/H\right] \ar[r] \ar[d] & \left[Y/G\right] \ar[d] \\
X \ar[r] & \left[X/H\right] \ar[r] & \left[X/G\right] \\
\end{tikzcd}$$
The proof of these claims follows from the same techniques as the ones used in Example <ref>. And for a concrete example, the group $\mu_n(\C)$ of $n$-th roots of unity acts (non-freely) on $\C$ by multiplication by $e^{i\frac{2\pi}{n}}$, and we have a covering stack (or stacky cover) $\C\lra[\C/\mu_n(\C)]$. The same construction works if we replace $\C$ by an open disk $D(0,\epsilon)\subset \C$.

Note that a covering stack is in particular an open morphism of stacks because the covering maps $f_M: \Y\times_\X M \lra M$ are all open morphisms in $\An$. We are now ready to give the definition of an orbifold. The intuition if that an analytic stack is an orbifold if it admits an open covering of the form $[U/\Gamma]$, where $U\in\An$ and $\Gamma$ is a finite group acting on $U$ by automorphisms.

An analytic orbifold is an analytic stack $\X$ (Definition <ref>) equipped with a family of substacks of the form $\big(F_i:[U_i/\Gamma_i]\lra \X\big)_{i\in I}$ such that:

* For all $i\in I$, the pair $(U_i,\Gamma_i)$ consists of a manifold $U_i\in\An$ and a finite group $\Gamma_i$ acting on $U_i$ by automorphisms.

* For all $i\in I$, the substack $F_i:[U_i/\Gamma_i]\lra \X$ is an open substack (Definition <ref>).

* The morphism of stacks $F:\sqcup_{i\in I} [U_i/\Gamma_i] \lra \X$ is surjective (Definition <ref>).

The stack $\sqcup_{i\in I} [U_i/\Gamma_i]$ appearing in Definition <ref> is defined, for all $S\in\An$, by $\big(\sqcup_{i\in I} [U_i/\Gamma_i]\big)(S) = \sqcup_{i\in I} \big([U_i/\Gamma_i](S)\big)$. An ultimately equivalent but slightly more abstract definition of an analytic orbifold would be that of an analytic stack equipped with a (surjective) finite cover $F:M\lra \X$ with $M\in\An$. For instance, all manifolds have canonical orbifold structures. Either way, the main source of examples is obtained as an application of the following result, which is due to Thurston in the case when the group $G$ acts effectively on the manifold $X$ <cit.>.

Let $X$ be an analytic manifold and let $G$ be a discrete group acting to the right on $X$ by automorphisms. For all $x\in X$, we denote by
$$G_x:=\{g\in G\ |\ x\cdot g = x\}$$
the stabiliser of $x$ in $G$. Assume that the following properties are satisfied:

* For all $x\in X$, there exists an open neighbourhood $U_x$ of $x$ such that

\begin{equation*}
g\in G_x \Rightarrow (U_x\cdot g) \subset U_x\quad \text{and}\quad g\notin G_x \Rightarrow (U_x \cdot g) \cap U_x = \emptyset.
\end{equation*}

* For all $x\in X$, the group $G_x$ is finite.

Then the quotient stack $[X/G]$ is an orbifold.

Since, for all $x\in X$, the subset $U_x\subset X$ is $G_x$-invariant and open in $X$, we know from Example <ref> that the quotient stack $[U_x/G_x]$ defines an open substack of $[X/G]$. And since the $(U_x)_{x\in X}$ form an open covering of $X$, the canonical morphism of stacks $F: \sqcup_{x\in X} [U_x/G_x] \lra [X/G]$ is surjective. Indeed, if $S\in\An$ and $(P,u)$ is a pair consisting of a principal $G$-bundle $P$ on $S$ and a $G$-equivariant map $u:P\lra X$, we can take an open covering $(S_i)_{i\in I}$ of $S$ such that, for all $i$, the principal $G$-bundle $P_i:=P|_{S_i}$ is trivial. Let us fix a trivialisation $P_i\simeq (U_i\times G)$ in which $u|_{P_i}:P_i\lra X$ is the action map, and choose $x\in X$ such that $x\in u(P_i)$. Set then $Q_{i,x} := (S_i \times G_x)$. By Condition (1), an element $\xi\in u^{-1}(U_x)\cap P_i$ satisfies $g\cdot\xi\in u^{-1}(U_x)\cap P_i$ if and only if $g\in G_x$, so $u^{-1}(U_x)\cap P_i \simeq Q_{i,x}$. So $P_i$ lies in the essential image of the canonical functor
$$F_{x,S_i}: [U_x/G_x](S_i) \lra [X/G](S_i),$$
hence also in the image of $F_S:\sqcup_{x\in X} [U_x/G_x](S_i)\lra [X/G](S_i)$, which proves that the morphism $F: \sqcup_{x\in X} [U_x/G_x]\lra [X/G]$ is surjective in the sense of Definition <ref>. Finally, since each $G_x$ is finite by Condition (2), we have indeed shown that $[X/G]$ is an orbifold.

If $G$ is finite to begin with, we can simply use the finite cover $X\lra[X/G]$ to prove that $[X/G]$ has a canonical orbifold structure. For instance, the quotient stack $[\C/\mu_n(\C)]$ of Example <ref> is canonically an orbifold. We will see in Corollary <ref> that the action of the discrete group $\SL(2;\Z)$ on the upper half-plane $\fh:=\{z\in\C\ |\ \text{Im}\,z>0\}$ satisfies the conditions of Theorem <ref>, so the quotient stack $[\fh/\SL(2;\Z)]$ is also an orbifold, in a canonical way.

Note that we are not assuming that the action of $G$ on $X$ is effective, only that it has finite stabilisers. In particular, if $G$ is a finite group, then the classifying stack $BG=[\bullet/G]$ is an orbifold. When the action of the group $G$ on the space $X$ is effective, the group $G$ can be identified with a subgroup of $\Aut(X)$ and the quotient stack $[X/G]$ is called an effective orbifold.orbifold!effective orbifold

To construct orbifolds, it suffices to find examples of actions of discrete groups on manifolds for which Conditions (1) and (2) of Theorem <ref> are satisfied. An important tool to do that is the following result. We state it for a real Lie group $H$, but we are mostly interested here in the case where $H$ is a Hermitian group, meaning that the symmetric space $X:=K\backslash H$ has a canonical structure of complex analytic (in fact, Kähler) manifold, in which case the quotient stack $[X/G]$ is indeed an analytic orbifold.

Let $H$ be a real Lie group whose Lie algebra admits an $H$-invariant positive definite scalar product. Fix a maximal compact subgroup $K\subset H$ and let us endow the homogeneous space $X:=K\backslash H$ with its canonical manifold structure. Let $G\subset H$ be a discrete subgroup of $H$. Then the canonical right action of $G$ on $X$ satisfies Conditions (1) and (2) of Theorem <ref>. In particular, the quotient stack $[X/G]$ has a canonical orbifold structure.

Since the upper half-plane $\fh=\{z\in\C\ |\ \mathrm{Im}\,z>0\}$ can be described as the homogeneous space $\mathbf{SO}(2)\backslash\SL(2;\R)$, the quotient stack $[\fh/\SL(2;\Z)]$ has a canonical orbifold structure.

Let us make a few remarks on the proof of Theorem <ref>. Remarks <ref> and <ref> show that proper, isometric actions of discrete subgroups of Lie groups on Riemannian manifolds give rise to orbifolds. Together with Remark <ref>, they can be formalised into a proof of Theorem <ref> and Corollary <ref>.

What is actually proven in <cit.> is that, if $G\subset H$ is a discrete subgroup of the Lie group $H$, the action of $G$ on $K\backslash H$ (or, as a matter of fact, on all manifold $X$ on which $H$ acts transitively with compact stabilisers $H_x$) satisfies the following condition:

(P) For all compact subset $C\subset X$, the set $\{g\in G\ |\ C\cdot g \cap C \neq \emptyset\}$ is finite.
An (effective) action of a group $G$ by homeomorphisms on a locally compact space $X$ is called properly discontinuousproperly discontinuous action if it satisfies Property (P); see <cit.>. If $G$ is endowed with the discrete topology, this is equivalent to asking that the continuous map $X\times G\lra X\times X$ given by $(x,g)\lmt (x,x\cdot g)$ be proper. So giving a properly discontinuous action on a locally compact space is equivalent to giving a proper, continuous action of a discrete group on that space <cit.>. Note that, for such an action, the stabilisers $G_x$ are finite, so Condition (2) of Theorem <ref> is indeed satisfied.

To prove Theorem <ref>, we still need to show that if a discrete group $G$ acts continuously and properly on a manifold $X$, then, as stated in the proof of <cit.>, for all $x\in X$, there exists an open neighbourhood $U_x$ of $x$ in $X$ which is $G_x$-invariant and disjoint from its translates by elements of $G$ that are not in $G_x$ (i.e. Condition (1) of Theorem <ref> is also satisfied). In fact, to prove Theorem <ref>, it suffices to do it in the case when there is a $G$-invariant distance $d$ on $X$ (i.e. the group $G$ acts by isometries on $X$). And in that case we can take $U_x$ to be a Dirichlet domainDirichlet domain
$$U_x = \mathcal{D}_G(x) := \{y\in X\ |\ \forall\,g\notin G_x, d(y,x)<d(y,x\cdot g)\}.$$
Such a set is $G_x$-invariant because, if $y\in \mathcal{D}_G(x)$ and $h\in G_x$, the $G$-invariance of $d$ gives, for all $g\notin G_x$,
$$d(y\cdot h, x\cdot g) = d(y, x\cdot gh^{-1}) > d(y,x) = d(y,x\cdot h^{-1}) = d(y\cdot h,x).$$
The set $\mathcal{D}_G(x)$ is also  disjoint from its translates by elements of $G$ that are not in $G_x$ because, if $g\notin G_x$, then we cannot have both $y\in\mathcal{D}_G(x)$ and $(y\cdot g)\in \mathcal{D}_G(x)$, as this would imply that
$$d(y,x) < d(y,x\cdot g^{-1}) = d(y\cdot g,x) < d(y\cdot g, x \cdot g) = d(y,x).$$
So there only remains to prove that $\mathcal{D}_G(x)$ is open. For the sake of completeness, we sketch here the arguments to be found for instance in <cit.>. Recall that $x$ is fixed throughout. For all $g\notin G_x$, set
$$H_g(x):=\{y\in X\ |\ d(y,x)<d(y,x\cdot g)\}\quad \text{and}\quad L_g(x) := \{y\in X\ |\ d(y,x) = d(y,x\cdot g)\}.$$
In particular, $H_g(x)$ is open, $L_g(x)$ is closed and $\mathcal{D}_G(x) = \cap_{g\notin G_x} H_g(x)$. Since $G$ is a discrete subgroup of a Lie group, it is countable. And since $G$ acts properly on $X$, a compact set $C\subset X$ can only intersect a finite number of $L_g(x)$. Indeed, this is a direct consequence of the fact that, if $G=\{g_0,g_1,\,\ldots\}$, then
$$\forall\,n\ |\ g_n\notin G_x, d\big(x,L_{g_n}(x)\big) = \textstyle{\frac{1}{2}} d\big(x,x\cdot g_n\big) \underset{n\to+\infty}{\lra} +\infty.$$
Now let us fix $y\in \mathcal{D}_G(x) = \cap_{g\notin G_x} H_g(x)$ and consider the compact disk $D[y;\epsilon]$ centred at $y$. As we have just seen, the set
$$\{g\notin G_x\ |\ D[y;\epsilon]\cap L_g(x)\not=\emptyset\} =: \{g_1,\,\ldots\,,g_r\}$$
is finite. Since $y\in H_{g}(x)$, we have that, for all $i\in\{1,\,\ldots\,,r\}$, there exists an integer $n_i\geq 1$ such that the open disk $D(y,\frac{\epsilon}{2^{n_i}})$ has an empty intersection with $L_{g_i}(x)$, and is therefore contained in $H_{g_i}(x)$. So, we can find an open neighbourhood $U_y := \cap_{i=1}^r D(y_i,\frac{\epsilon}{2^{n_i}})$ such that, for all $g\notin G_x$, $U_y\subset H_g(x)$, i.e. $U_y\subset \mathcal{D}_G(x)$, proving that $\mathcal{D}_G(x)$ is indeed open.

Note that, in the situation of Theorem <ref>, a $G$-invariant distance exists on $X=K\backslash H$ because $H$ possesses a left-invariant Riemannian metric. Indeed we have assumed that the Lie algebra of $H$ admits an $H$-invariant positive definite scalar product (this holds for instance if $H$ is a real form of a reductive complex Lie group, such as $H=\SL(2;\R)$ in Corollary <ref>). Note that the complex analytic structure of $\fh$ indeed comes from its description as the homogeneous space $\mathbf{SO}(2)\backslash \SL(2;\R)$ because of the exceptional isomorphism between $\SL(2;\R)$ and the Hermitian real form $\mathbf{SU}(1,1)$ of $\SL(2;\C)$.

As explained after the proof of Corollary 3.5.11 in <cit.>, if $X$ is a space of the type considered in Theorem <ref>, Condition (P) in Remark <ref> admits equivalent characterisations, all of them given in <cit.>. For instance the fact that $G$ is a discrete subgroup of $H$, or the fact that the action is wandering, in the following sense:

(W) For all $x\in X$, there is an open neighbourhood $U_x$ of $x$ such that the set $\{g\in G\ |\ U_x\cdot g \cap U_x \neq \emptyset\}$ is finite.
If $X$ is locally compact, then (P) $\Rightarrow$ (W). And if $X$ is a metric space such that $G$ acts on $X$ by isometries, then (W) $\Rightarrow$ (P). We shall not need this notion in what follows.

§ MODULI SPACES

§.§ Coarse moduli spaces and fine moduli spaces

It is sometimes desirable to approximate an analytic stack by a space, in an appropriately optimal sense that we now define.

moduli space!coarse moduli space
Let $\X$ be an analytic stack. A coarse moduli space for $\X$ is a pair $(M,\pi)$ where $M\in\An$ and $\pi:\X\lra M$ is a morphism of stacks satisfying the following universal property:

$$\text{For\ all\ morphism\ of\ stacks}\ \phi:\X\lra N\ \text{with}\ N\in\An, \exists!\, \overline{\phi}:M\lra N\ |\ \overline{\phi}\circ\pi=\phi.$$

Note that the morphism $\overline{\phi}:M\lra N$ is a morphism of stacks between two manifolds, so by the Yoneda lemma <ref> it is induced by a uniquely defined morphism of analytic manifolds. The condition that $(M,\pi)$ be a coarse moduli space for $\X$ is usually summed up by the commutativity of the following diagram.

\begin{tikzcd}
\X \ar[r, "\phi"] \ar[d, "\pi"] & N\\
M \ar[ru, "\exists!\,\overline{\phi}"', dashed] &
\end{tikzcd}

Sometimes, an extra condition is added, namely the fact that the functor $\pi_\bullet:\X(\bullet)\lra M(\bullet)\simeq M$ be surjective, or in other words, that there be a surjective map from the set $\X(\bullet)/\negmedspace\sim$ of isomorphism classes of objects parameterised by the stack $\X$ to the underlying set of the manifold $M$. Note however that asking for this map to be bijective (as opposed to just surjective) might be too restrictive (insofar as, in some examples, the universal property and the surjectivity condition are satisfied, but not the bijectivity condition). Typically, this happens when $\X=[X/G]$ and the action of $G$ on $X$ has non-closed orbits (see Example <ref>). To understand this, we first need to study the existence of coarse moduli spaces for quotient stacks. The key observation is the following result.

Let $G$ be a complex Lie group acting to the right on an analytic manifold $X$. Then, the canonical morphism $p:X\lra[X/G]$ defines a bijection $\phi\lmt \phi\circ p$ between morphisms of stacks $\phi$ from $[X/G]$ to a manifold $N\in\An$ and $G$-invariant morphisms $\psi:X\lra N$ in $\An$:

\begin{equation*}
\left\{
\text{morphisms\ of\ stacks}\
\phi:[X/G]\lra N
\right\}
\quad \simeq \quad
\left\{
G\text{-invariant\ morphisms}\ \psi:X\lra N\ \text{in}\ \An
\right\}
\end{equation*}
More precisely, given a $G$-invariant morphism $\psi:X\lra N$ in $\An$, there is a unique morphism of stacks $\phi:[X/G]\lra N$ such that $\phi\circ p = \psi$.

Proposition <ref> says that the quotient stack $[X/G]$ satisfies the following universal property with respect to $G$-invariant morphisms $ \psi: X \lra N$.

\begin{tikzcd}
X \ar[r, "\psi"] \ar[d, "p"] & N\\
\left[X/G\right] \ar[ru, "\exists!\,\phi"', dashed] &
\end{tikzcd}

First note that, as in Example <ref>, we have a $2$-Cartesian diagram

\begin{equation}\label{pullback_diag_for_triv_bdle}
\begin{tikzcd}
X \times_{\left[X/G\right]} X \simeq X\times G \ar[r,"\text{action}"] \ar[d, "pr_1"] & X\ar[d, "p"] \\
X \ar[r, "p"'] & \left[X/G\right]
\end{tikzcd}
\end{equation}
whose commutativity can be taken as a definition that the morphism of stacks $p:X\lra[X/G]$ is $G$-invariant. The commutativity property also implies that, if $\phi:[X/G]\lra N$ is a morphism of stacks whose target stack is a manifold $N\in\An$, then the morphism $\psi:=\phi\circ p:X\lra N$ is $G$-invariant as a morphism in $\An$. Now assume that a $G$-invariant morphism $\psi:X\lra N$ has been given in $\An$. We will show that, conversely, there is a unique morphism of stacks $\phi:[X/G]\lra N$ such that $\phi\circ p=\psi$, .i.e. that the universal property depicted in Diagram (<ref>) holds for the quotient stack $[X/G]$.

\begin{equation}\label{univ_ppty_of_quotient_stacks_diagram}
\begin{tikzcd}
X\times G \ar[r,"\text{action}"] \ar[d, "pr_1"] & X\ar[d, "p"] \ar[rdd, bend left=20, "\psi"]& \\
X \ar[r, "p"'] \ar[rrd, bend right=20, "\psi"] & \left[X/G\right] \ar[dr, dashed, "\exists!\,\phi"'] & \\
& & N
\end{tikzcd}
\end{equation}

Recall that to define $\phi$ is to define $\phi_S:[X/G](S)\lra \ul{N}(S)$ for all $S\in\An$. But an object in $[X/G](S)$ is a pair $(P,u)$ consisting of a principal $G$-bundle $P$ on $S$ and a $G$-equivariant map $u:P\lra X$. So we have a $G$-invariant morphism $\psi\circ u:P\lra N$ in $\An$. Since $P$ is a principal $G$-bundle on $S$, the morphism $\psi\circ u$ factors into a unique morphism $f:S\simeq P/G\lra N$ (by the universal property of the quotient $P/G$).

\begin{tikzcd}
P \ar[d] \ar[r, "u"] & X \ar[r,"\psi"] & N\\
S \ar[rru, dashed, "\exists\,!f"'] & &
\end{tikzcd}

The morphism $\phi_S:[X/G](S)\lra \ul{N}(S)$ is then defined by $\phi_S(P,u) := f$.

As an immediate consequence of Proposition <ref>, we get the following special case of Definition <ref>: for quotient stacks, the notion of coarse moduli space coincides with the classical notion of categorical quotient. In particular, depending on the action, they may or may not exist.

quotient!categorical quotient
Let $G$ be a complex Lie group acting to the right on a complex analytic manifold $X$. The quotient stack $[X/G]$ admits a coarse moduli space if and only if the action of $G$ on $X$ admits a categorical quotient, i.e. if there exists a pair $(M,\rho)$ where $M\in\An$ and $\rho:X\lra M$ is a morphism of analytic manifolds such that:

* The map $\rho$ is $G$-invariant: for all $g\in G$ and all $x\in X$, $\rho(x\cdot g) = \pi(x)$.

* For all $G$-invariant morphism of analytic manifolds $\psi: X \lra N$, there exists a unique morphism of analytic manifolds $\overline{\psi}:M\lra N$ such that the following diagram is commutative:

\begin{equation}\label{univ_ppty_quot_space}
\begin{tikzcd}
X \ar[r, "\psi"] \ar[d, "\rho"] & N\\
M \ar[ru, "\exists!\,\overline{\psi}"', dashed] &
\end{tikzcd}
\end{equation}

A particularly good case is when $M=X/G$, i.e. when the orbit space $X/G$ is a categorical quotient. This happens when the orbit space satisfies the conditions of the following definition.

quotient!geometric quotient
Let $G$ be a complex Lie group acting to the right on an analytic manifold $X$ and denote by $ \rho : X \lra X/G $ the canonical projection. If the orbit space $X/G$, endowed with the quotient topology, admits a structure of complex analytic manifold with respect to which the holomorphic functions on an open subset $ U \subset X/G $ correspond, via pullback by $\rho$, to $ G $-invariant holomorphic functions on the open subset $\rho^{-1}(U) \subset X $, then $X/G$ is called a geometric quotient for the $ G $-action on $ X $.

It can be checked that a geometric quotient is a categorical quotient: it necessarily satisfies the universal property (<ref>). An example of a situation in which a geometric quotient exists is the case when $G$ acts freely and properly on $ X $ (see Example <ref>). Note that, since the points of $X/G$ are closed and the canonical projection $\rho:X\lra X/G$ is continuous, a geometric quotient can only exist if the $G$-orbits in $X$ are closed, and if there are non-closed orbits, then the points of a categorical quotient (if it exists) cannot be in bijection with the orbits of the action (see Example <ref>).  There are cases in which the orbits are closed and the action is not free but a geometric quotient exists: this is in fact what happens for the action of the modular group $\SL(2;\Z)$ on the upper half-plane $ \fh $ (see also Example <ref>).

If a complex Lie group $ G $ acts freely and properly on a complex analytic manifold $ X $, then the orbit space $ X / G $ admits a structure of complex analytic manifold that turns the canonical projection $ \rho : X \lra X/G $ into a holomorphic principal $ G $-bundle. In particular, the map $ \rho $ has local sections and holomorphic functions on $ X / G $ do indeed correspond to $ G $-invariant holomorphic functions on $ X $, so $ X / G$ is indeed a geometric quotient when $ G $ acts freely and properly on $ X $. As a matter of fact, we will see that more is true in that case: when the action is proper and free, the orbit space $ X / G $ is a fine moduli space for the quotient stack $[X/G]$ (see Definition <ref> and Proposition <ref>).

Let $ G = \C^* $ act on $ X = \C^2 $ via $ t \cdot ( z_1, z_2 )  := ( t \cdot z_1, t^{-1} \cdot z_2 ) $. The closed orbits of this action are the origin $\{(0,0)\}$, which is a fixed point of the action, and the hyperbolas of equation $ z_1 z_2 = \lambda $, for all $ \lambda\in\C^*$. The origin is contained in the closure of the only two non-closed orbits, which are the real and imaginary axes minus the origin. In particular, a $G$-invariant morphism $\psi: X \lra N$ will take the same value on these three orbits, which will therefore be identified in the categorical quotient. This gives us the intuition that the categorical quotient in this example should be the set of closed orbits of the $G$-action, which is in bijection with $\C$ via the $ G $-invariant function $\rho: \C^2 \lra \C$ sending $( z_1, z_2 )$ to $z_1z_2$. By construction, this set has strictly less elements than the whole set of orbits of the $ G $-action. Sometimes, it can even have drastically less elements: if instead of the above action, we let $ G = \C^* $ act on $ X = \C^2 $ via $ t \cdot ( z_1, z_2 )  = ( t \cdot z_1, t \cdot z_2 ) $, then the only closed orbit is $\{(0,0)\}$, which gives us the intuition that the categorical quotient will be a point in this case.

In Section <ref>, we will give sufficient conditions for the existence of categorical and geometric quotients for actions of reductive complex Lie groups. As we shall see, in certain cases, a geometric quotient $X/G$ can exist even if the action of $ G $ on $ X $ is not free: a simple example of this is given in Example <ref>. We end this section with the notion of fine moduli space, which is the case when the stack $\chi$ is not only approximated by a manifold $M$, but actually represented by it.

moduli space!fine moduli space
Let $\X$ be an analytic stack. A fine moduli space for $\X$ is a complex analytic manifold $M$ such that $ \X \simeq M$.

It is readily checked that a fine moduli space is also a coarse moduli space for $ \X $. As an example, let us study the existence of fine moduli spaces for a quotient stack $\X = [X/G]$.

Let $G$ be a complex analytic group acting to the right on a complex analytic manifold $X$. The quotient stack $[X/G]$ admits a fine moduli space if and only if $G$ acts freely and properly on $ G $. In that case, the canonical projection $p:X \lra X/G$ defines a holomorphic principal $G$-bundle.

Let us first assume that $[X/G]$ admits a fine moduli space $M$. Then the canonical morphism $X \lra [X/G]\simeq M$ is a principal $G$-bundle, so $G$ acts freely and properly on $X$ and $M\simeq X/G$. Conversely, if $G$ acts freely and properly on $X$, let us consider the principal $G$-bundle $X \lra X/G$. For all $S\in\An$, an object in $[X/G](S)$ is a principal $G$-bundle $P\lra S$ equipped with a $G$-equivariant map $u:P\lra X$. Since $X/G$ is a manifold, the $G$-equivariant morphism $u$ induces a morphism of complex analytic manifolds $\hat{u}:P/G\simeq S\lra X/G\simeq M$. Conversely, such a map $f:S\lra M$ induces, by pullback of the principal $G$-bundle $X\lra X/G\simeq M$, a principal $G$-bundle $f^*X\lra S$, equipped by construction with a $G$-equivariant map $P\lra X$. These two maps set up natural isomorphisms $[X/G](S) \simeq M(S)$, hereby proving that $[X/G]$ is representable by $M$.

The action of $\C^*$ on $\C^2\setminus\{(0,0)\} $ given by $t \cdot ( z_1, z_2 )  := ( t \cdot z_1, t \cdot z_2 ) $ is proper and free. So the quotient stack $[ \, (\C^2\setminus\{(0,0)\}) \, /  \, \C^*]$ admits $\CP^1$ as its fine moduli space.

We can reformulate Proposition <ref> by saying that, for quotient stacks, a fine moduli space exists if and only if the action is free and admits a geometric quotient.

§.§ GIT quotients

To make sense out of Example <ref> from a general point of view, we can use the Kempf-Ness formulation of Geometric Invariant Theory ([7, 8]). The work of Kempf and Ness applies to analytic actions of reductive complex Lie groups $ G $ (i.e. groups that are isomorphic to the complexification of their maximal compact subgroup) and, at least when $ X $ can be embedded equivariantly and linearly onto a closed submanifold of some $ \C^N $ (meaning $G$-equivariantly with respect to a $ G $-action on $ \C^N$ that is induced by a linear representation $ G \lra \GL(N;\C)$), it enables us to say us two things:

* If $ O \subset X $ is a $ G $-orbit, then its closure $ \overline{O} $ contains a unique closed orbit. As a consequence, the relation $ O_1 \sim O_2 $ if $ \overline{O_1} \cap \overline{O_2} \neq \emptyset $ is an equivalence relation on the set of $ G $-orbits (the previous result on the closure of an orbit being used to prove that this relation is transitive).

* The set of closed $ G $-orbits in $ X $, denoted by $ X /\!\!/ G $, which by definition is the quotient of $ X / G $ by the previous equivalence relation, is a categorical quotient for the $ G $-action on $ X $. In particular, the holomorphic functions on $X/\!\!/G$ are the $G$-invariant holomorphic functions on $ X $. This quotient is called the GIT quotient for the action of $ G $ on $ X $.

There is in fact a more general situation, in which $ X $ can be embedded $ G$-equivariantly onto a locally closed submanifold of some $ \CP^{N-1}$ with respect to a linearisable action of $G$, meaning that $ G $ is required to act on $ \CP^{N-1} $ via a a linear representation $ G \lra \GL(N;\C)$ and the canonical action of $ \GL(N;\C)$ on $\CP^{N-1}$. In that case, one has to restrict the action to the so-called semistable locus $ X^{ss} \subset X $, which is a $G$-invariant open subset of $ X $ that can be defined as follows:

$$ X^{ss} = \{ x \in X \subset \CP^{N-1}\ | \ 0 \not\in \overline{G\cdot \tilde{x}}, \text{ where } \tilde{x}\in \C^N \text{ is a lift of } x\}. $$
If $X^{ss}\neq\emptyset$, then one can look at $ G $-orbits that are closed in $ X^{ss} $ and proceed as before to form the categorical quotient $ X^{ss} /\!\!/G $. The result, however, can depend strongly on the linearisation (see Example <ref>). In practice, it suffices to test semistability with respect to all $1$-parameter subgroups $\lambda: \C^* \lra G$, which is done via the Hilbert-Mumford criterion:

$$ X^{ss} = \big\{ x \in X \subset \CP^{N-1}\ | \ \forall\, \lambda: \C^* \lra G, \ \exists\, \mu \geq 0, \ \lim_{t \to 0} t^\mu\, \big( \lambda(t)\cdot \tilde{x} \big) \text{ exists in } \C^N \text{ and is} \neq 0 \big\} . $$

Let us equip $\C^2$ with the $\C^*$-action given by $t \cdot (z_1,z_2) = (tz_1,tz_2)$, and identify it with an open subset $ X \subset \CP^2$ via $(z_1,z_2) \lmt [1:z_1,z_2]$, then:

* With respect to the linearisation $t\cdot[z_0:z_1:z_2] := [tz_0:tz_1:tz_2]$, it can be checked (via the Hilbert-Mumford criterion) that $X^{ss} = \emptyset$, so there is no categorical quotient in this case.

* With respect to the so-called trivial linearisation $t\cdot[z_0:z_1:z_2] := [z_0:tz_1:tz_2]$, the semistable locus is
$$X^{ss} =  \big\{ [ z_0 : z_1 : z_2] \in \CP^2 \ | \ z_0\neq 0 \big\} = X \simeq \C^2.$$
The only closed orbit is $\{(0,0)\}$ so the categorical quotient is a point in this case.

* With respect to the linearisation $t\cdot[z_0:z_1:z_2] := [t^{-1}z_0:tz_1:tz_2]$, the semistable locus is

$$ X^{ss} = \big\{ [ z_0 : z_1 : z_2] \in \CP^2 \ | \ z_0\neq 0 \text{ and } (z_1,z_2) \neq (0,0) \big\} \simeq \C^2 \setminus
\big\{ (0,0) \big\} \, .$$
All $\C^*$-orbits are then closed in $X^{ss}$, so in this case the categorical quotient is

$$ X^{ss} /\!\!/ \C^* \simeq  \big(\C^2\setminus\{(0,0)\}\big) \, /  \, \C^* \simeq \CP^1 .$$

For a presentation of geometric invariant theory in a complex analytic setting (using the results of Kempf and Ness), we refer for instance to [18], the point being that, via that theory, linearisable actions of reductive complex Lie groups on quasi-projective analytic manifolds give rise to special categorical quotients called GIT quotientsquotient!GIT quotient. More precisely, we have the following result.

Let $ G $ be a reductive complex Lie group acting linearly on a quasi-projective analytic manifold $ X \subset \CP^{N-1} $. If $ X^{ss} \neq \emptyset $, then the GIT quotient

$$ X^{ss} /\!\!/ G := \{ \text{closed } G\text{-orbits in } X^{ss} \}$$

is a coarse moduli space for the analytic stack $[X^{ss}/G]$.

Geometric invariant theory tells us that, as $ G $ is reductive, the space of closed semistable orbits $X^{ss} /\!\!/G$ carries a structure of complex analytic manifold that makes it a categorical quotient for the action of $ G $ on $ X^{ss}$. By Corollary <ref>, the GIT quotient $X^{ss} /\!\!/ G $ is therefore a coarse moduli space for the analytic stack $[X^{ss}/G]$.

When a reductive group $ G $ acts on a locally closed analytic submanifold $X \subset \C^N$, we can embed $ X $ equivariantly in $\CP^{N-1}$ with respect to the trivial linearization $ g \cdot [z_0: w] = [z_0 : g\cdot w] $, where $w\in\C^N$. Then $X^{ss} = X$, so in that case the quotient stack $[X/G]$ admits a coarse moduli space.

Let us retake the example of the orbifold $[\C/\mu_n(\C)]$, where the group $ \mu_n(\C) \simeq \Z/n\Z $ of $n$-th roots of unity acts on $\C$ via $ \zeta \cdot z = \zeta z $ (Examples <ref> and <ref>). This action is proper but not free. Since $\mu_n(\C)$ is a finite group, it is reductive. So we get a GIT quotient whose orbits are all closed, and we note that such a GIT quotient is necessarily a geometric quotient in the sense of Definition <ref>, because by construction the holomorphic functions on a GIT quotient $X^{ss} /\!\!/ G$ correspond to $ G $-invariant functions on $ X^{ss} $. In the present example, we can identify this geometric quotient with $\C$ as a complex analytic manifold, via the holomorphic map $ j : \C/\mu_n(\C) \lra \C $ induced by the $\mu_n(\C)$-invariant map $z \lmt z^n$ (see Diagram <ref>). The same occurs if we replace $ \C $ by an open disk centred at the origin.

\begin{equation}\label{CMS_roots_of_unity_acting_by_rotation}
\begin{tikzcd}
\C \ar[rd, "z\lmt z^n"] \ar[d, "\rho"'] & \\
\C/\mu_n(\C) \ar[r, "j"'] & \C
\end{tikzcd}
\end{equation}

To sum up, when $ G $ is a reductive complex analytic Lie group acting linearly on a quasi-projective complex analytic manifold $ X $ with non-empty semistable locus $X^{ss}$, GIT quotients $X^{ss} /\!\!/ G$ exist in $\An$ and satisfy the following properties:

\text{GIT quotient} \Rightarrow \text{coarse moduli space/categorical quotient for the quotient stack} \left[X^{ss}/G\right] $$

$$ \text{GIT quotient + closed orbits} \Rightarrow \text{geometric quotient for the quotient stack} \left[X^{ss}/G\right].

By definition, the points of the GIT quotient $X^{ss} /\!\!/ G$ are the closed $ G $-orbits in $X^{ss}$, so saying that all orbits are closed is exactly saying that $X^{ss} /\!\!/ G= X^{ss} / G$ (i.e. that the GIT quotient is an orbit space). Finally:

\text{GIT quotient + closed orbits + free action} \Rightarrow \text{fine moduli space for the quotient stack } \left[X^{ss}/G\right] .

§ THE MODULI STACK OF ELLIPTIC CURVES

§.§ Families of elliptic curves

Given a complex analytic manifold $S\in\An$, we define $\M_{1,1}(S)$ to be the category of analytic families of elliptic curves parameterised by $S$, in the following sense.

family!of elliptic curves
A family of elliptic curves parameterised by a complex analytic manifold $ S \in \An $ is a triple $(E,\pi,e)$ where:

* $E$ is a complex analytic manifold.

* $\pi:E\lra S$ is a submersion in $\An$.

* $e:S\lra E$ is a section of $\pi$.

* For all $s\in S$, the pair $(E_s:=\pi^{-1}(\{s\}),e(s))$ is a complex elliptic curve, i.e. a compact connected Riemann surface $E_s$ of genus $1$, equipped with a marked point $e(s)\in E_s$. In particular, the map $\pi$ is surjective.
We define a morphism of families from $(E_1,\pi_1,e_1)$ to $(E_2,\pi_2,e_2)$ to be an isomorphism $u:E_1\lra E_2$ in $\An$ such that $\pi_2\circ u = \pi_1$ and $u\circ e_1 = e_2$.

\begin{equation*}
\begin{tikzcd}
E_1 \ar[rr, "u"] \ar[dr, "\pi_1"] & & E_2 \ar[dl, "\pi_2"'] \\
& S \ar[ur, dashed, bend right=20, "e_2"'] \ar[lu, dashed, bend left=20, "e_1"] &
\end{tikzcd}
\end{equation*}

The Legendre family $E = \sqcup_{u\in \C\setminus\{0,1\}} \sC_u $ defined in Example <ref> is a family of elliptic curves in the sense of Definition <ref>, parameterised by $S:=\C\setminus\{0,1\}$. Indeed,

$$ E := \left\{ (u, [x:y:z]) \in S \times \mathbb{CP}^{2} \ | \ y^{2}z=x(x-z)(x-u z)\right\}$$
is a complex analytic submanifold of $S \times \mathbb{CP}^{2} $, mapping to $ S =  \C \setminus\{0,1\} $ via the projection $\pi$ to the first factor, which is a submersion. The fibre of $\pi: E \lra S$ above $u\in S$ is the genus $1$ Riemann surface $\sC_u$ constructed in Example <ref>, and the points $[0:0:1]\in\sC_u$ define a global section $e: s \lmt (s, [0:0:1])$ of $ \pi$.

We shall often drop $\pi$ from the notation and simply write $(E,e)$ for a family of elliptic curves parameterised by $S$. Since we have restricted ourselves to isomorphisms between analytic families, the category $\M_{1,1}(S)$ is a groupoid, for all $ S \in \An $. To check that this defines a prestack
$$ \M_{1,1}:\An^{\mathrm{op}} \lra \text{Groupoids} $$
in the sense of Definition <ref>, we must check that, for all morphism $f:T\lra S$ in $\An$, we can define a natural pullback functor $f^{\,*}:\M_{1,1}(S)\lra \M_{1,1}(T)$. To do this, we set, for all $(E,\pi,e)\in \M_{1,1}(S)$:

\begin{equation}\label{pullback_curve}
f^*E := \{(x,t)\in E\times T\ |\ \pi(x) = f(t)\}
\end{equation}
Since $\pi:E\lra S$ is a holomorphic submersion, the topological space $f^*E$ admits a structure of complex analytic manifold, turning the canonical projection $f^*E\lra T$ into a submersion. Moreover, for all $t\in T$, the fibre $(f^*E)_t \simeq E_{f(t)}$ has a canonical structure of complex elliptic curve, with marked point $e(f(t))=(f^*e)(t)$. This proves that $(f^*E,f^*e)$ is an analytic family of elliptic curves parametrised by $T$ and that $f^*E$ if the fibre product of $E$ and $T$ over $S$ in $\An$. So $E\lmt f^*E$ is a functor with respect to $E$ and, if $g:U\lra T$ is a morphism in $\An$, we can identify canonically $g^*(f^*E)$ and $(f\circ g)^*E$ in $\M_{1,1}(U)$, which is what we need for Definition <ref>.

In the general definition of a family of complex analytic manifolds, the condition that the holomorphic submersion $\pi:E\lra S$ also be proper is often incorporated in the definition. In particular, with that definition, the fibres of $\pi$ are automatically compact.

moduli stack!of elliptic curvesstack!moduli stack
The prestack $\M_{1,1}$ defined, for all complex analytic manifold $S\in\An$, by
$$\M_{1,1}(S) = \big\{\text{analytic families\ of\ elliptic\ curves\ parameterised\ by}\ S\big\}$$
is called the moduli stack of elliptic curves.

The term stack used in Definition <ref> is justified by the following result.

The prestack $\M_{1,1}$ is a stack.

We have to check that the conditions of Definition <ref> are satisfied by the prestack $\M_{1,1}$. This means showing that analytic families of elliptic curves parameterised by $S\in\An$, and morphisms between them, can be constructed by gluing. Explicitly, we want to show that, given an analytic manifold $ S $ and an open covering $ ( S_i )_{ i \in I } $ of $ S $, the following two properties hold:

* If $ E, E' \lra S $ are families of elliptic curves parameterised by $ S $ and $ \phi_i : E|_{ S_i } \lra E'|_{ S_i } $ are morphisms between the pullbacks of these two families to $ S_i $, satisfying the compatibility conditions  $ \phi_i|_{ S_i \cap S_j } = \phi_j|_{ S_i \cap S_j } $ over $ S_i \cap S_j $, then there exists a unique morphism $ \phi : E \lra E ' $ in $ \M_{1,1}( S ) $ such that, for all $ i $, $ \phi|_{ S_i } = \phi_i $.

* If, for all $ i $ in $ I $, we are given a family of elliptic curves $ \pi_i : E_i \lra S_i $, as well as isomorphisms
$$\phi_{ij}: E_j|_{S_i\cap S_j} \overset{\simeq}{\lra} E_i|_{S_i\cap S_j}$$
satisfying the cocycle conditions $\phi_{ii} = \id_{E_i}$ and $\phi_{ij}\circ\phi_{jk} = \phi_{ik}$ over $ S_i \cap S_j \cap S_k$, then there exists a family of elliptic curves $ E \lra S $ together with isomorphisms $\phi_i:E|_{S_i}\lra E_i$ over $ S_i $, such that, for all $i,j$ in $ I $, one has $ \phi_i \circ \phi_j^{-1} = \phi_{ij} $ over $ S_i \cap S_j $.

The first property holds because we can simply define $ \phi( x ) := \phi_i( x ) $ if $ x \in E|_{ S_i } $ and this gives a well-defined map that is a morphism of families of elliptic curves from $ E $ to $ E' $, with the property that $ \phi|_{ S_i } = \phi_i $. And the second property holds because the complex analytic manifold

$$ E := \sqcup_{i \in I} E_i \, \big/ \sim $$
where $( i, x) \sim (j, y)$ if $ x = \phi_{ij}( y ) $ satisfies the required properties by construction: it is equipped with a holomorphic submersion

\begin{equation}\label{hol_submersion_constructed_by_gluing}
\pi : E \lra \left(\sqcup_{i \in I} S_i \, \big/ \sim\right) = S
\end{equation}
whose fibres are elliptic curves because the gluing procedure guarantees that $ E $ comes equipped with isomorphisms between the open subsets $ E|_{S_i} \subset E $ and the family $ E_i $.

Note that if we had asked, in Definition <ref>, that the holomorphic submersion $ \pi: E \lra S $ also be proper, Property (2) in the proof of Proposition <ref> would be more difficult to show: we would also need to prove that the map (<ref>), constructed by gluing, is proper, which is in fact the case when the morphisms $\pi_i: E_i \lra S_i$, from which $ \pi : E \lra S $ is constructed, are also assumed to be proper.

We ultimately want to argue that the reason why the stack $ \M_{1,1} $ is called a moduli stack is because of the existence of a universal familyfamily!universal family on it. This will require an explanation, but the idea is that there exists, because of the geometric content behind the definition of $ \M_{1,1} $, a family of elliptic curves $ \U_{1,1} \lra \M_{1,1}$ with the following property:

\begin{equation}\label{univ_ppty_of_univ_family}
\begin{matrix}
\text{For all } S \in \An \text{ and all } E \in \M_{1,1}(S) \text{, there exists} \text{ a unique}\\ \text{morphism } f: S \lra \M_{1,1}  \text{ such that } f^* \U_{1,1} \simeq E .
\end{matrix}
\end{equation}

Note that, for Property (<ref>) to make sense, we must provide a definition of a family of elliptic curves parameterised by a stack instead of a manifold. Once this is done, we can get an intuition of how we can define a universal family.

Let $\N$ be a stack over $\An$. A family of elliptic curvesfamily!of elliptic curves over $\N$ is a morphism of stacks $\chi: \E \lra \N $ such that:

* The morphism $\chi: \E \lra \N$ is representable.

* For all $S \in \An$ and all morphism $f : S \lra \N$, the pullback
$$ f^*\E \simeq S \times_\N \E \in \An $$
is a family of elliptic curves over $ S $, in the sense of Definition <ref>.

In Definition <ref>, the role of Condition (1) is to ensure that Condition (2) makes sense. Indeed, Condition (1) guarantees that $f^*\E$ is a manifold for all  morphism $f : S \lra \N$ with $S \in \An$.

In view of Definition <ref>, we can extend $\M_{1,1}$ to the category of stacks by defining groupoids

\begin{equation}\label{families_param_by_a_stack}
\widehat{\M_{1,1}}(\N) := \{ \E \lra \N : \ \text{family of elliptic curves on} \ \N \}\,.
\end{equation}

Note that a family of elliptic curves $\E\lra \N$ induces a morphism of stacks $F: \N \lra \M_{1,1}$, defined, for all $ S \in \An $, by taking $(u:S \lra \N)$ to $u^*\E \in \M_{1,1}(S)$. For this to be compatible with the Yoneda embedding $\M_{1,1} \lmt \Mor_{\mathcal{S}t}(-,\M_{1,1})$, it is necessary and sufficient to assume that the the identity morphism $\id_{\M_{1,1}}$ comes, in the same fashion, from a family $\U_{1,1} \lra \M_{1,1}$. If this is the case, then by the same argument as in Example <ref>, the family $\U_{1,1}$ must indeed satisfy the universal property (<ref>). So in order to define a universal family of elliptic curves, it suffices to be able to interpret $\id_{\M_{1,1}}$ as a family of elliptic curves over the stack $\M_{1,1}$.

The preceding discussion suggests that all stacks can be abstractly thought of as moduli stacks: if $\M$ is a stack on $\An$, we can consider the $2$-functor $\Mor_{\mathcal{S}t}(-,\M) : \mathcal{S}t/\An \lra \text{Groupoids}$ and define an extension of $\M:\An \lra \text{Groupoids}$ to $\widehat{\M}: \mathcal{S}t/\An \lra \text{Groupoids}$ by setting $\widehat{\M}(\N) := \Mor_{\mathcal{S}t}(-,\M)$ for all $\N\in\mathcal{S}t/\An$. The role of the universal family $\N\in\widehat{\M}(\M)$ is then played by the morphism $\id_{\M}$. But ultimately we want to have a more concrete (and geometrically interpretable) description of universal families, which will depend on the precise stack $\M$ being studied.

When $\M=\M_{1,1}$, we will describe the universal family of elliptic curves in Corollary <ref>. For the moment, we content ourselves with two elementary but fundamental examples of universal families.

Let $G$ be a complex analytic group and let $BG:= [\bullet/G]$ be the classifying stack of principal $G$-bundles. By Example <ref>, for all manifold $S \in \An$, the groupoid $BG(S)$ is the   category of principal $G$-bundles on $X$, and by Example <ref>, the stack $BG$ is an analytic stack. The universal family is then the morphism $EG:=\bullet \lra BG = [ \bullet / G]$. Indeed, it is a representable morphism by Example <ref> and, by Example <ref>, it satisfies the universal property (<ref>). This means that, for all $S\in \An$ and all principal $G$-bundle $P$ on $S$, there exists a unique morphism of stacks $f:S\lra BG$ such that $P\simeq f^* EG$. Because of this universal property, the principal $G$-bundle $EG\lra BG$ is called the universal bundlebundle!universal bundle over the classifying stack $BG$. Note that $EG$ is indeed a principal $G$-bundlebundle!principal bundle on a stack over the stack $BG$, where this notion is to be taken in a sense similar to the covering stacks of Definition <ref>: a representable morphism $EG\lra BG$ whose pullback to a manifold $S\in\An$ along a morphism of stacks $S \lra BG$ is a principal $G$-bundle in $\An$. Also, comments similar to Proposition <ref> and Remark <ref> apply: since $BG$ is an analytic stack, to check that the representable morphism $EG \lra BG$ is a principal $G$-bundle, it suffices to check it with respect to the atlas $EG \lra BG$, over which the pullback $EG$ is just $EG \times G$, the trivial principal $G$-bundle over $EG=\bullet$, as seen by taking $X=\bullet$ in Diagram (<ref>).

Note that $BG$ is not a representable stack as soon a $G$ is non-trivial, but that the universal family $EG=\bullet$ is (representable by) a manifold.

As a generalisation of Example <ref>, consider the principal $G$-bundle $X \lra [X/G]$, where $G$ is a complex analytic group acting on a complex analytic manifold $X$. To say that this is the universal family over the stack $[X/G]$ defined in Example <ref> is to say that, for all $S\in\An$ and all object $(P,u)\in [X/G](S)$, there exists a unique morphism of stacks $f: S \lra [X/G]$ such that $P \simeq f^*X$ and that the $G$-equivariant morphism $f^*X \lra X$ coincides with $u: P \lra X$. In other words, it means finding an $f: S \lra [X/G]$ such that the diagram

\begin{tikzcd}
P \ar[r,"u"'] \ar[d] & X\ar[d] \\
S \ar[r, "f"'] & \left[X/G\right]
\end{tikzcd}
is a pullback diagram. This is possible because the principal $G$-bundle $P$ is locally trivial over $S$ and we have already noted that, if $P = X\times G$ and $u: X\times G \lra X$ is the action map, then  Diagram (<ref>) is a pullback diagram.

§.§ Families of framed elliptic curves

We now wish to study the representability of the moduli stack $\M_{1,1}$. As we shall see, this stack can be seen not to be representable for rather general and abstract reasons, which have to do with the fact that the objects that it parameterises, namely elliptic curves, have non-trivial automorphisms (Corollary <ref>) and this prevents the moduli stack of elliptic curves from being representable (Proposition <ref>). Since one can get rid of automorphisms by adding a framing to elliptic curves (Theorem <ref>), a natural route to study $\M_{1,1}$ is to introduce a moduli stack of framed elliptic curvesmoduli stack!of framed elliptic curves $\M_{1,1}^{\,\mathrm{fr}}$ and relate it to $\M_{1,1}$. We will ultimately show that $\M_{1,1}^{\,\mathrm{fr}}$, in contrast to $\M_{1,1}$, is representable (by the Teichmüller space $\fh$) and this will imply that the moduli stack $\M_{1,1}$ is isomorphic to the quotient stack $[\fh/\SL(2;\Z)]$. This last step is what shows that $\M_{1,1}$ is an orbifold in the sense of Definition <ref> (in particular, an analytic stack with atlas $\fh$) and that it admits the modular curve $\fh/\SL(2;\Z)$ as a coarse moduli space. But before we can get to all this, let us make a couple of remarks.

Let us assume that there exists a family of elliptic curves $\U_{1,1} \lra \M_{1,1}$ (in the sense of Definition <ref>) such that the associated morphism $\M_{1,1} \lra \M_{1,1}$ (defined, for all $S\in\An$, by sending a morphism of stacks $f: S \lra \M_{1,1}$ to the family of elliptic curves $f^*\U_{1,1} \in \M_{1,1}(S)$) is the identity morphism. Then, as we have noted, the stack $\U_{1,1}$ is a universal family in the sense that it satisfies the universal property (<ref>). As suggested by Example <ref>, the representability of the universal family need not imply the representability of the moduli stack. In the reverse direction, however, the representability of $\M_{1,1}$ (by a manifold $M_{1,1}$, say) would imply that the universal family $\U_{1,1}$ is representable (by the fibre product $M_{1,1}\times_{\M_{1,1}} \U_{1,1}$, which is indeed a manifold because the morphism of stacks $\U_{1,1} \lra \M_{1,1}$ is by assumption representable).

More generally, if we can prove that $\M_{1,1}$ is an analytic stack, then the universal family $\U_{1,1}$ (which we have not yet constructed), is automatically an analytic stack. Indeed, since $\U_{1,1}\lra\M_{1,1}$ is representable, the fibre product of $\U_{1,1}$ with an atlas $X \lra \M$ gives a manifold $X\times_{\M_{1,1}} \U_{1,1}$, which is an atlas for $\U_{1,1}$ because it is straightforward to check that all morphisms in the pullback diagram

\begin{tikzcd}
X\times_{\M_{1,1}} \U_{1,1} \ar[r] \ar[d] & \U_{1,1}\ar[d] \\
X \ar[r] & \M_{1,1}
\end{tikzcd}

are representable surjective submersions.

Independently of these remarks, we can prove the following result.

The moduli stack of elliptic curves $\M_{1,1}$ is not representable by a manifold.

Let us assume that the stack $\M_{1,1}$ is representable by a manifold $M_{1,1}\in\An$, i.e. that $\M_{1,1} \simeq \Mor_{\An}(-,M_{1,1})$. The contradiction will come from a special property of representable functors such as $\Mor_{\An}(-,M_{1,1})$, namely that if $f: X \lra Y$ is a covering map in $\An$, then the induced map

\begin{equation}\label{pullback_by_cover}
f^*: \Mor_{\An}(Y,M_{1,1}) \lra \Mor_{\An}(X,M_{1,1})
\end{equation}
is injective. In fact, it suffices to know that this property holds for Galois covers $ Y = X / G$ where $G$ is a finite group acting by automorphisms on $Y$. In this case, the image of the pullback map (<ref>) is the set $\mathrm{Fix}_G\, \Mor_{\An}(X,M_{1,1})$ of $G$-invariant complex analytic maps from $X$ to $M_{1,1}$ and since such a map induces a well-defined complex analytic map from $Y$ to $M_{1,1}$, the pullback map $f^*$ is injective.

Thus, to prove that $\M_{1,1}$ is not representable, it suffices to find a Galois covering map $\pi: X \lra X/G$ and two non-isomorphic families of elliptic curves $E_1, E_2$ on $X/G$ such that $\pi^*E_1 \simeq \pi^* E_2$ as families of elliptic curves over $X$. Indeed, this will contradict the injectivity of $\pi^*: \M_{1,1}(Y) \lra \M_{1,1}(X)$ which has to hold if we assume that $\M_{1,1} \simeq \Mor_{\An}(-,M_{1,1})$, i.e. that $\M_{1,1}$ is representable. And this is when the fact that there are elliptic curves with non-trivial automorphisms comes in handy (to construct the families $E_1, E_2$).

Let $\sC$ be the elliptic curve $\sC := \C / (\Z1 \oplus \Z i)$ and set $X := \sC$. We consider the action of the group $G := \Z/2\Z$ induced on $E := X\times \sC$ by the involution

\begin{equation}\label{concrete_invol}
\big([x], [z]\big) \lmt \big( [x+\textstyle\frac{1}{2}], [-z] \big)
\end{equation}
and the quotient $E_1 := E / G$. Since $G$ acts freely on $X$, the quotient $Y:= X/G$ is a Riemann surface and the canonical projection $\pi:X \lra Y$ is a two-to-one cover. Note that $Y$ has genus $1$ but does not carry a preferred marked point because $G$ does not act on $X$ by automorphisms of elliptic curves. Moreover, there is an induced complex analytic map $E_1 \lra Y$, which is locally trivial with fibres isomorphic to $\sC$. Since the action of $G$ on the fibres of $E$ is an action by automorphisms of elliptic curves, the map $E_1 \lra Y$ is in fact a family of elliptic curves. The point is that $E_1$ is not trivial but pulls back to the trivial family $E = X \times \sC$. Since the product family like $E_2 := Y \times \sC$ also pulls back to $E$, we see that the map $\pi^*: \M_{1,1}(Y) \lra \M_{1,1}(X)$ is not injective.

Note that, to see that $E_1$ is not trivial, it suffices to show that it does not carry a non-vanishing holomorphic volume form (which a product of two Riemann surfaces of genus $1$ would), because otherwise the pullback of such a volume form to $E = X\times \sC$ would be $G$-invariant and equal to $c dx \wedge dz $ for some constant $c\neq 0$, contradicting the fact that $dx \wedge dz $ is $G$-anti-invariant with respect to the involution (<ref>).

The technique used in the proof of Proposition <ref> can be adapted to show the non-representability of various moduli stacks. It rests on the fact that a representable stack $\M \simeq \Mor_{\An}(-,M)$ is a sheaf in various Grothendieck topologies on $\An$, in particular the étale topology. So, to prove that a stack $\M$ is not representable, it suffices to find an étale covering $\pi: Y\lra X$ such that $\pi^*:\M(X) \lra \M(Y)$ is not  injective. Indeed, this injectivity is part of the conditions of being a sheaf: the restriction map (or pullback) to an open covering (or more generally an étale cover) is an injective map (if two sections of a sheaf coincide on every open of the cover, they are equal). Then, in order to show a lack of injectivity, the usual technique (as in the proof of Proposition <ref>) consists in finding a non-trivial family whose pullback to a finite cover is trivial. The construction of such a family (called an isotrivial familyfamily!isotrivial family) is then typically obtained by taking the quotient of a trivial family by a finite group acting by automorphisms on the fibre and freely on the base.