In mathematics, the moduli stack of elliptic curves, denoted as ${\displaystyle {\mathcal {M))_{1,1))$ or ${\displaystyle {\mathcal {M))_{\textrm {ell))}$, is an algebraic stack over ${\displaystyle {\text{Spec))(\mathbb {Z} )}$ classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves ${\displaystyle {\mathcal {M))_{g,n))$. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme ${\displaystyle S}$ to it correspond to elliptic curves over ${\displaystyle S}$. 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 ${\displaystyle {\mathcal {M))_{1,1))$.

Properties

Smooth Deligne-Mumford stack

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

j-invariant

There is a proper morphism of ${\displaystyle {\mathcal {M))_{1,1))$ 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 ${\displaystyle \mathbb {C} }$ is classified by its periods. Given a basis for its integral homology ${\displaystyle \alpha ,\beta \in H_{1}(E,\mathbb {Z} )}$ and a global holomorphic differential form ${\displaystyle \omega \in \Gamma (E,\Omega _{E}^{1})}$ (which exists since it is smooth and the dimension of the space of such differentials is equal to the genus, 1), the integrals

$${\displaystyle {\begin{bmatrix}\int _{\alpha }\omega &\int _{\beta }\omega \end{bmatrix))={\begin{bmatrix}\omega _{1}&\omega _{2}\end{bmatrix))}$$

${\displaystyle \mathbb {Z} }$

${\displaystyle \mathbb {C} }$

${\displaystyle \Lambda }$

${\displaystyle 2}$

${\displaystyle \mathbb {C} }$

${\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda }$

${\displaystyle \mathbb {P} ^{2))$

${\displaystyle \phi :\mathbb {C} /\Lambda \to E(\mathbb {C} )}$

$${\displaystyle z\mapsto [\wp (z,\Lambda ),\wp '(z,\Lambda ),1]\in \mathbb {P} ^{2}(\mathbb {C} )}$$

${\displaystyle \Lambda }$

$${\displaystyle z\Lambda \sim \Lambda ~{\text{for))~z\in \mathbb {C} \setminus \{0\))$$

${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau }$

${\displaystyle \tau \in {\mathfrak {h))}$

${\displaystyle \Lambda }$

${\displaystyle \omega _{1}^{-1))$

${\displaystyle \tau ,-\tau }$

${\displaystyle \mathbb {C} }$

$${\displaystyle {\text{SL))_{2}(\mathbb {Z} )=\left\((\begin{pmatrix}a&b\\c&d\end{pmatrix))\in {\text{Mat))_{2,2}(\mathbb {Z} ):ad-bc=1\right\))$$

${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau }$

${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau '}$

$${\displaystyle {\begin{aligned}{\begin{pmatrix}a&b\\c&d\end{pmatrix))\cdot \tau &={\frac {a\tau +b}{c\tau +d))\\&=\tau '\end{aligned))}$$

${\displaystyle \mathbb {C} }$

$${\displaystyle {\mathcal {M))_{1,1}\cong [{\text{SL))_{2}(\mathbb {Z} )\backslash {\mathfrak {h))]}$$

${\displaystyle {\text{PSL))_{2}(\mathbb {Z} )={\text{SL))_{2}(\mathbb {Z} )/\{\pm I\))$

${\displaystyle {\mathcal {M))_{1,1))$

${\displaystyle \qquad }$

Stacky/Orbifold points

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

Representing involutions of plane curves

Given a plane curve by its Weierstrass equation

$${\displaystyle y^{2}=x^{3}+ax+b}$$

${\displaystyle (t,s)}$

${\displaystyle j\neq 0,1728}$

${\displaystyle \mathbb {Z} /2}$

${\displaystyle (t,s)\mapsto (t,-s)}$

$${\displaystyle y^{2}=x^{3}+ax}$$

${\displaystyle \mathbb {Z} /4}$

${\displaystyle (t,s)\mapsto (-t,{\sqrt {-1))\cdot s)}$

${\displaystyle a=0}$

$${\displaystyle y^{2}=x^{3}+b}$$

${\displaystyle \mathbb {Z} /6}$

${\displaystyle (t,s)\mapsto (\zeta _{3}t,-s)}$

${\displaystyle \zeta _{3))$

${\displaystyle e^{2\pi i/3))$

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

$${\displaystyle D=\{z\in {\mathfrak {h)):|z|\geq 1{\text{ and )){\text{Re))(z)\leq 1/2\))$$

${\displaystyle {\mathcal {M))_{1,1))$

$${\displaystyle {\mathfrak {h))\to {\text{SL))_{2}(\mathbb {Z} )\backslash {\mathfrak {h))}$$

${\displaystyle D}$

${\displaystyle {\text{Re))(z)\mapsto -{\text{Re))(z)}$

${\displaystyle {\mathcal {M))_{1,1))$

${\displaystyle \mathbb {P} ^{1))$

Line bundles and modular functions

There are line bundles ${\displaystyle {\mathcal {L))^{\otimes k))$ over the moduli stack ${\displaystyle {\mathcal {M))_{1,1))$ whose sections correspond to modular functions ${\displaystyle f}$ on the upper-half plane ${\displaystyle {\mathfrak {h))}$. On ${\displaystyle \mathbb {C} \times {\mathfrak {h))}$ there are ${\displaystyle {\text{SL))_{2}(\mathbb {Z} )}$-actions compatible with the action on ${\displaystyle {\mathfrak {h))}$ given by

$${\displaystyle {\text{SL))_{2}(\mathbb {Z} )\times {\displaystyle \mathbb {C} \times {\mathfrak {h))}\to {\displaystyle \mathbb {C} \times {\mathfrak {h))))$$

${\displaystyle k}$

$${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix)):(z,\tau )\mapsto \left((c\tau +d)^{k}z,{\frac {a\tau +b}{c\tau +d))\right)}$$

${\displaystyle \mathbb {C} \times {\mathfrak {h))\to {\mathfrak {h))}$

${\displaystyle k}$

${\displaystyle {\mathcal {L))^{\otimes k))$

${\displaystyle \mathbb {C} }$

${\displaystyle {\text{SL))_{2}(\mathbb {Z} )}$

${\displaystyle \mathbb {Z} }$

${\displaystyle {\mathcal {L))^{\otimes k}\otimes {\mathcal {L))^{\otimes l}\cong {\mathcal {L))^{\otimes (k+l)))$

${\displaystyle {\mathcal {L))^{\otimes k))$

${\displaystyle f\in \Gamma (\mathbb {C} \times {\mathfrak {h)))}$

${\displaystyle {\text{SL))_{2}(\mathbb {Z} )}$

${\displaystyle f:{\mathfrak {h))\to \mathbb {C} }$

$${\displaystyle f\left({\begin{pmatrix}a&b\\c&d\end{pmatrix))\cdot \tau \right)=(c\tau +d)^{k}f(\tau )}$$

Modular forms

The modular forms are the modular functions which can be extended to the compactification

$${\displaystyle {\overline ((\mathcal {L))^{\otimes k))}\to {\overline {\mathcal {M))}_{1,1))$$

${\displaystyle {\mathcal {M))_{1,1))$

${\displaystyle q}$

${\displaystyle q}$

Universal curves

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

$${\displaystyle [({\text{SL))_{2}(\mathbb {Z} )\ltimes \mathbb {Z} ^{2})\backslash \mathbb {C} \times {\mathfrak {h))]}$$

Versal curve

Every rank 2 ${\displaystyle \mathbb {Z} }$-lattice in ${\displaystyle \mathbb {C} }$ induces a canonical ${\displaystyle \mathbb {Z} ^{2))$-action on ${\displaystyle \mathbb {C} }$. As before, since every lattice is homothetic to a lattice of the form ${\displaystyle (1,\tau )}$ then the action ${\displaystyle (m,n)}$ sends a point ${\displaystyle z\in \mathbb {C} }$ to

$${\displaystyle (m,n)\cdot z\mapsto z+m\cdot 1+n\cdot \tau }$$

${\displaystyle \tau }$

${\displaystyle {\mathfrak {h))}$

${\displaystyle \mathbb {Z} ^{2))$

${\displaystyle \mathbb {C} \times {\mathfrak {h))}$

$${\displaystyle (m,n)\cdot (z,\tau )\mapsto (z+m\cdot 1+n\cdot \tau ,\tau )}$$

$${\displaystyle {\mathcal {E))_{\mathfrak {h))\to {\mathfrak {h))}$$

${\displaystyle {\mathfrak {h))}$

SL₂-action on Z²

There is a ${\displaystyle {\text{SL))_{2}(\mathbb {Z} )}$-action on ${\displaystyle \mathbb {Z} ^{2))$ which is compatible with the action on ${\displaystyle {\mathfrak {h))}$, meaning given a point ${\displaystyle z\in {\mathfrak {h))}$ and a ${\displaystyle g\in {\text{SL))_{2}(\mathbb {Z} )}$, the new lattice ${\displaystyle g\cdot z}$ and an induced action from ${\displaystyle \mathbb {Z} ^{2}\cdot g}$, which behaves as expected. This action is given by

$${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix)):(m,n)\mapsto (m,n)\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix))}$$

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