# Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as ${\mathcal {M}}_{1,1}$ or ${\mathcal {M}}_{\mathrm {ell} }$, 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 ${\mathcal {M}}_{1,1}$.

## Properties
### 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 $\omega \in \Gamma (E,\Omega _{E}^{1})$ (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 $\mathbb {C}$ $^{pg158}$. 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 $^{pg165}$. 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${\begin{aligned}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot \tau &={\frac {a\tau +b}{c\tau +d}}\\&=\tau '\end{aligned}}$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 $^{pg10-11}$ corresponding to elliptic curves with $j$-invariant equal to $1728$ and $0$ where the automorphism groups are of order 4, 6, respectively $^{pg170}$. 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}$$^{pg78}$.

#### 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$^{pg78}$. 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$^{pg52}$.

### 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${\begin{pmatrix}a&b\\c&d\end{pmatrix}}:(z,\tau )\mapsto \left((c\tau +d)^{k}z,{\frac {a\tau +b}{c\tau +d}}\right)$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$f\left({\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot \tau \right)=(c\tau +d)^{k}f(\tau )$ 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)$^{pgs29-33}$.

### 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 ${\mathfrak {h}}$.

#### SL$_2$-action on Z$^2$

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$(m,n)\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=(am+cn,bm+dn)$