Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as $$\mathcal{M}_{1,1}$$ or $$\mathcal{M}_{\textrm{ell}}$$, is an algebraic stack over $$\text{Spec}(\mathbb{Z})$$ classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves $$\mathcal{M}_{g,n}$$. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme $$S$$ to it correspond to elliptic curves over $$S$$. The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in $$\mathcal{M}_{1,1}$$.

Smooth Deligne-Mumford stack
The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over $$\text{Spec}(\mathbb{Z})$$, but is not a scheme as elliptic curves have non-trivial automorphisms.

j-invariant
There is a proper morphism of $$\mathcal{M}_{1,1}$$ to the affine line, the coarse moduli space of elliptic curves, given by the j-invariant of an elliptic curve.

Construction over the complex numbers
It is a classical observation that every elliptic curve over $$\mathbb{C}$$ is classified by its periods. Given a basis for its integral homology $$\alpha,\beta \in H_1(E,\mathbb{Z})$$ and a global holomorphic differential form $$\omega \in \Gamma(E,\Omega^1_E)$$ (which exists since it is smooth and the dimension of the space of such differentials is equal to the genus, 1), the integrals$$\begin{bmatrix}\int_\alpha \omega & \int_\beta\omega \end{bmatrix} = \begin{bmatrix}\omega_1 & \omega_2 \end{bmatrix}$$give the generators for a $$\mathbb{Z}$$-lattice of rank 2 inside of $$\mathbb{C}$$ pg 158. Conversely, given an integral lattice $$\Lambda$$ of rank $$2$$ inside of $$\mathbb{C}$$, there is an embedding of the complex torus $$E_\Lambda = \mathbb{C}/\Lambda$$ into $$\mathbb{P}^2$$ from the Weierstrass P function pg 165. This isomorphic correspondence $$\phi:\mathbb{C}/\Lambda \to E(\mathbb{C})$$ is given by$$z \mapsto [\wp(z,\Lambda),\wp'(z,\Lambda),1] \in \mathbb{P}^2(\mathbb{C})$$and holds up to homothety of the lattice $$\Lambda$$, which is the equivalence relation$$z\Lambda \sim \Lambda ~\text{for}~ z \in \mathbb{C} \setminus\{0\}$$It is standard to then write the lattice in the form $$\mathbb{Z}\oplus\mathbb{Z}\cdot \tau$$ for $$\tau \in \mathfrak{h}$$, an element of the upper half-plane, since the lattice $$\Lambda$$ could be multiplied by $$\omega_1^{-1}$$, and $$\tau,-\tau$$ both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over $$\mathbb{C}$$. There is an additional equivalence of curves given by the action of the$$\text{SL}_2(\mathbb{Z})= \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{Mat}_{2,2}(\mathbb{Z}) : ad-bc = 1 \right\}$$where an elliptic curve defined by the lattice $$\mathbb{Z}\oplus\mathbb{Z}\cdot \tau$$ is isomorphic to curves defined by the lattice $$\mathbb{Z}\oplus\mathbb{Z}\cdot \tau'$$ given by the modular action$$ \begin{align} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \tau &= \frac{a\tau + b}{c\tau + d} \\ &= \tau' \end{align}$$Then, the moduli stack of elliptic curves over $$\mathbb{C}$$ is given by the stack quotient$$ \mathcal{M}_{1,1} \cong[\text{SL}_2(\mathbb{Z})\backslash\mathfrak{h}]$$Note some authors construct this moduli space by instead using the action of the Modular group $$\text{PSL}_2(\mathbb{Z}) = \text{SL}_2(\mathbb{Z})/\{\pm I\}$$. In this case, the points in $$\mathcal{M}_{1,1}$$ having only trivial stabilizers are dense.



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Stacky/Orbifold points
Generically, the points in $$\mathcal{M}_{1,1}$$ are isomorphic to the classifying stack $$B(\mathbb{Z}/2)$$ since every elliptic curve corresponds to a double cover of $$\mathbb{P}^1$$, so the $$\mathbb{Z}/2$$-action on the point corresponds to the involution of these two branches of the covering. There are a few special points pg 10-11 corresponding to elliptic curves with $j$-invariant equal to $$1728$$ and $$0$$ where the automorphism groups are of order 4, 6, respectively pg 170. One point in the Fundamental domain with stabilizer of order $$4$$ corresponds to $$\tau = i$$, and the points corresponding to the stabilizer of order $$6$$ correspond to $$\tau = e^{2\pi i / 3}, e^{\pi i / 3}$$ pg 78.

Representing involutions of plane curves
Given a plane curve by its Weierstrass equation$$y^2 = x^3 + ax + b$$and a solution $$(t,s)$$, generically for j-invariant $$j \neq 0,1728$$, there is the $$\mathbb{Z}/2$$-involution sending $$(t,s)\mapsto (t,-s)$$. In the special case of a curve with complex multiplication$$ y^2 = x^3 + ax$$there the $$\mathbb{Z}/4$$-involution sending $$(t,s)\mapsto (-t,\sqrt{-1}\cdot s)$$. The other special case is when $$a = 0$$, so a curve of the form$$y^2 = x^3 + b$$ there is the $$\mathbb{Z}/6$$-involution sending $$(t,s) \mapsto (\zeta_3 t,-s)$$ where $$\zeta_3$$ is the third root of unity $$e^{2\pi i / 3}$$.

Fundamental domain and visualization
There is a subset of the upper-half plane called the Fundamental domain which contains every isomorphism class of elliptic curves. It is the subset$$D = \{z \in \mathfrak{h} : |z| \geq 1 \text{ and } \text{Re}(z) \leq 1/2 \}$$It is useful to consider this space because it helps visualize the stack $$\mathcal{M}_{1,1}$$. From the quotient map$$\mathfrak{h} \to \text{SL}_2(\mathbb{Z})\backslash \mathfrak{h}$$the image of $$D$$ is surjective and its interior is injective pg 78. Also, the points on the boundary can be identified with their mirror image under the involution sending $$\text{Re}(z) \mapsto -\text{Re}(z)$$, so $$\mathcal{M}_{1,1}$$ can be visualized as the projective curve $$\mathbb{P}^1$$ with a point removed at infinity pg 52.

Line bundles and modular functions
There are line bundles $$\mathcal{L}^{\otimes k}$$ over the moduli stack $$\mathcal{M}_{1,1}$$ whose sections correspond to modular functions $$f$$ on the upper-half plane $$\mathfrak{h}$$. On $$\mathbb{C}\times\mathfrak{h}$$ there are $$\text{SL}_2(\mathbb{Z})$$-actions compatible with the action on $$\mathfrak{h}$$ given by$$\text{SL}_2(\mathbb{Z}) \times {\displaystyle \mathbb {C} \times {\mathfrak {h}}} \to {\displaystyle \mathbb {C} \times {\mathfrak {h}}}$$The degree $$k$$ action is given by$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} : (z,\tau ) \mapsto \left( (c\tau + d)^kz, \frac{a\tau + b}{c\tau + d} \right)$$hence the trivial line bundle $$\mathbb{C}\times\mathfrak{h} \to \mathfrak{h}$$ with the degree $$k$$ action descends to a unique line bundle denoted $$\mathcal{L}^{\otimes k}$$. Notice the action on the factor $$\mathbb{C}$$ is a representation of $$\text{SL}_2(\mathbb{Z})$$ on $$\mathbb{Z}$$ hence such representations can be tensored together, showing $$\mathcal{L}^{\otimes k} \otimes \mathcal{L}^{\otimes l} \cong \mathcal{L}^{\otimes (k + l)}$$. The sections of $$\mathcal{L}^{\otimes k}$$ are then functions sections $$f \in \Gamma(\mathbb{C}\times \mathfrak{h})$$ compatible with the action of $$\text{SL}_2(\mathbb{Z})$$, or equivalently, functions $$f:\mathfrak{h} \to \mathbb{C}$$ such that$$ f\left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot \tau \right) = (c\tau + d)^kf(\tau)$$ This is exactly the condition for a holomorphic function to be modular.

Modular forms
The modular forms are the modular functions which can be extended to the compactification$$\overline{\mathcal{L}^{\otimes k}} \to \overline{\mathcal{M}}_{1,1}$$this is because in order to compactify the stack $$\mathcal{M}_{1,1}$$, a point at infinity must be added, which is done through a gluing process by gluing the $$q$$-disk (where a modular function has its $$q$$-expansion) pgs 29-33.

Universal curves
Constructing the universal curves $$\mathcal{E} \to \mathcal{M}_{1,1}$$ is a two step process: (1) construct a versal curve $$\mathcal{E}_{\mathfrak{h}} \to \mathfrak{h}$$ and then (2) show this behaves well with respect to the $$\text{SL}_2(\mathbb{Z})$$-action on $$\mathfrak{h}$$. Combining these two actions together yields the quotient stack$$[(\text{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2 )\backslash \mathbb{C}\times\mathfrak{h}]$$

Versal curve
Every rank 2 $$\mathbb{Z}$$-lattice in $$\mathbb{C}$$ induces a canonical $$\mathbb{Z}^{2}$$-action on $$\mathbb{C}$$. As before, since every lattice is homothetic to a lattice of the form $$(1,\tau)$$ then the action $$(m,n)$$ sends a point $$z \in \mathbb{C}$$ to$$(m ,n)\cdot z \mapsto z + m\cdot 1 + n\cdot\tau$$Because the $$\tau$$ in $$ \mathfrak{h}$$ can vary in this action, there is an induced $$\mathbb{Z}^{2}$$-action on $$\mathbb{C}\times\mathfrak{h}$$$$(m ,n)\cdot (z, \tau) \mapsto (z + m\cdot 1 + n\cdot\tau, \tau)$$giving the quotient space$$\mathcal{E}_\mathfrak{h} \to \mathfrak{h}$$by projecting onto $$\mathfrak{h}$$.

SL2-action on Z2
There is a $$\text{SL}_2(\mathbb{Z})$$-action on $$\mathbb{Z}^{2}$$ which is compatible with the action on $$\mathfrak{h}$$, meaning given a point $$z \in \mathfrak{h}$$ and a $$g \in \text{SL}_2(\mathbb{Z})$$, the new lattice $$g\cdot z$$ and an induced action from $$\mathbb{Z}^2 \cdot g$$, which behaves as expected. This action is given by$$\begin{pmatrix} a & b \\ c & d \end{pmatrix} : (m, n) \mapsto (m,n)\cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$which is matrix multiplication on the right, so$$(m,n)\cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ( am + cn, bm + dn )$$