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 $\omega \in \Gamma(E,\Omega^1_E)$ (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 $\mathbb{C}$ {{Cite book |last=Silverman, Joseph H. |title=The arithmetic of elliptic curves |date=2009 |publisher=Springer-Verlag |isbn=978-0-387-09494-6 |edition=2nd |location=New York |oclc=405546184}} ^{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.

File:The_modular_group_PSL_2(Z).svg

$\qquad$

= 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{{cite arXiv|last=Hain|first=Richard|date=2014-03-25|title=Lectures on Moduli Spaces of Elliptic Curves|class=math.AG|eprint=0812.1803}} ^{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{{Cite book |last=Galbraith |first=Steven |chapter=Elliptic Curves |chapter-url=https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf |title=Mathematics of Public Key Cryptography |url=https://www.math.auckland.ac.nz/~sgal018/crypto-book/crypto-book.html |publisher=Cambridge University Press |via=The University of Auckland}} ^{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 $\tau = e^{2\pi i / 3}, e^{\pi i / 3}$ {{Cite book |last=Serre, Jean-Pierre |title=A Course in Arithmetic |date=1973 |publisher=Springer New York |isbn=978-1-4684-9884-4 |location=New York |oclc=853266550}}^{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{{Cite book |author=Henriques, André G |chapter=The Moduli stack of elliptic curves |url=https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf |title=Topological modular forms |editor=Douglas, Christopher L. |editor2=Francis, John |editor3=Henriques, André G |editor4=Hill, Michael A. |isbn=978-1-4704-1884-7|location=Providence, Rhode Island |oclc=884782304|archive-url=https://web.archive.org/web/20200609190825/https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf|archive-date=9 June 2020 |via=University of California, Los Angeles}}^{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 $\mathfrak{h}$ .

== SL<sub>2</sub>-action on Z<sup>2</sup> ==

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

References

{{citation|title=Lectures on Moduli Spaces of Elliptic Curves|first=Richard|last= Hain|arxiv=0812.1803|year=2008|bibcode=2008arXiv0812.1803H}}
{{citation|last=Lurie|first= Jacob |year=2009|url=http://www.math.harvard.edu/~lurie/papers/survey.pdf |title=A survey of elliptic cohomology}}
{{Citation | last1=Olsson | first1=Martin | title=Algebraic spaces and stacks |year=2016|publisher=American Mathematical Society|series=Colloquium Publications|volume=62|isbn=978-1470427986}}

External links

{{nlab|id=moduli+stack+of+elliptic+curves}}
{{citation|chapter-url=http://stacks.math.columbia.edu/tag/072K|chapter=The moduli stack of elliptic curves|title=Stacks project}}

Category:Algebraic geometry