Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as $\mathcal_$ or $\mathcal_$, is an algebraic stack over $\text(\mathbb)$ classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves $\mathcal_$. 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 $\mathcal_$.

Properties

Smooth Deligne-Mumford stack

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

j-invariant

There is a proper morphism of $\mathcal_$ 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 $\mathbb$ is classified by its periods. Given a basis for its integral homology $\alpha,\beta \in H_1(E,\mathbb)$ 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$\begin\int_\alpha \omega & \int_\beta\omega \end = \begin\omega_1 & \omega_2 \end$give the generators for a $\mathbb$-lattice of rank 2 inside of $\mathbb$ ^{pg 158}. Conversely, given an integral lattice $\Lambda$ of rank $2$ inside of $\mathbb$, there is an embedding of the complex torus $E_\Lambda = \mathbb/\Lambda$ into $\mathbb^2$ from the Weierstrass P function ^{pg 165}. This isomorphic correspondence $\phi:\mathbb/\Lambda \to E(\mathbb)$ is given byz \mapsto wp(z,\Lambda),\wp'(z,\Lambda),1\in \mathbb^2(\mathbb)and holds up to homothety of the lattice $\Lambda$, which is the equivalence relation$z\Lambda \sim \Lambda ~\text~ z \in \mathbb \setminus\$It is standard to then write the lattice in the form $\mathbb\oplus\mathbb\cdot \tau$ for $\tau \in \mathfrak$, an element of the upper half-plane, since the lattice $\Lambda$ could be multiplied by $\omega_1^$, and $\tau,-\tau$ both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over $\mathbb$. There is an additional equivalence of curves given by the action of the$\text_2(\mathbb)= \left\$where an elliptic curve defined by the lattice $\mathbb\oplus\mathbb\cdot \tau$ is isomorphic to curves defined by the lattice $\mathbb\oplus\mathbb\cdot \tau'$ given by the modular action$\begin \begin a & b \\ c & d \end \cdot \tau &= \frac \\ &= \tau' \end$Then, the moduli stack of elliptic curves over $\mathbb$ is given by the stack quotient \mathcal_ \cong text_2(\mathbb)\backslash\mathfrak/math>Note some authors construct this moduli space by instead using the action of the Modular group $\text_2(\mathbb) = \text_2(\mathbb)/\$. In this case, the points in $\mathcal_$ having only trivial stabilizers are dense.

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Stacky/Orbifold points

Generically, the points in $\mathcal_$ are isomorphic to the classifying stack $B(\mathbb/2)$ since every elliptic curve corresponds to a double cover of $\mathbb^1$, so the $\mathbb/2$-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 $\tau = e^, e^$^{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 $\mathbb/2$-involution sending $(t,s)\mapsto (t,-s)$. In the special case of a curve with complex multiplication$y^2 = x^3 + ax$there the $\mathbb/4$-involution sending $(t,s)\mapsto (-t,\sqrt\cdot s)$. The other special case is when $a = 0$, so a curve of the form$y^2 = x^3 + b$ there is the $\mathbb/6$-involution sending $(t,s) \mapsto (\zeta_3 t,-s)$ where $\zeta_3$ is the third root of unity $e^$.

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 = \$It is useful to consider this space because it helps visualize the stack $\mathcal_$. From the quotient map$\mathfrak \to \text_2(\mathbb)\backslash \mathfrak$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 $\text(z) \mapsto -\text(z)$, so $\mathcal_$ can be visualized as the projective curve $\mathbb^1$ with a point removed at infinity^{pg 52}.

Line bundles and modular functions

There are line bundles $\mathcal^$ over the moduli stack $\mathcal_$ whose sections correspond to modular functions $f$ on the upper-half plane $\mathfrak$. On $\mathbb\times\mathfrak$ there are $\text_2(\mathbb)$-actions compatible with the action on $\mathfrak$ given by$\text_2(\mathbb) \times \to$The degree $k$ action is given by$\begin a & b \\ c & d \end : (z,\tau ) \mapsto \left( (c\tau + d)^kz, \frac \right)$hence the trivial line bundle $\mathbb\times\mathfrak \to \mathfrak$ with the degree $k$ action descends to a unique line bundle denoted $\mathcal^$. Notice the action on the factor $\mathbb$ is a representation of $\text_2(\mathbb)$ on $\mathbb$ hence such representations can be tensored together, showing $\mathcal^ \otimes \mathcal^ \cong \mathcal^$. The sections of $\mathcal^$ are then functions sections $f \in \Gamma(\mathbb\times \mathfrak)$ compatible with the action of $\text_2(\mathbb)$, or equivalently, functions $f:\mathfrak \to \mathbb$ such that$f\left( \begin a & b \\ c & d \end \cdot \tau \right) = (c\tau + d)^kf(\tau)$ 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$\overline \to \overline_$this is because in order to compactify the stack $\mathcal_$, 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 $\mathcal \to \mathcal_$ is a two step process: (1) construct a versal curve $\mathcal_ \to \mathfrak$ and then (2) show this behaves well with respect to the $\text_2(\mathbb)$-action on $\mathfrak$. Combining these two actions together yields the quotient stack \text_2(\mathbb) \ltimes \mathbb^2 )\backslash \mathbb\times\mathfrak/math>

Versal curve

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

SL₂-action on Z²

There is a $\text_2(\mathbb)$-action on $\mathbb^$ which is compatible with the action on $\mathfrak$, meaning given a point $z \in \mathfrak$ and a $g \in \text_2(\mathbb)$, the new lattice $g\cdot z$ and an induced action from $\mathbb^2 \cdot g$, which behaves as expected. This action is given by$\begin a & b \\ c & d \end : (m, n) \mapsto (m,n)\cdot \begin a & b \\ c & d \end$which is matrix multiplication on the right, so$(m,n)\cdot \begin a & b \\ c & d \end = ( am + cn, bm + dn )$

See also

* Fundamental domain
* Homothety
* Level structure (algebraic geometry)
* Moduli of abelian varieties
* Shimura variety
* Modular curve
* Elliptic cohomology

References

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External links

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*{{citation, chapter-url=http://stacks.math.columbia.edu/tag/072K, chapter=The moduli stack of elliptic curves, title=Stacks project

Algebraic geometry