Moduli stack of elliptic curves

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

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

give the generators for a ${\displaystyle \mathbb {Z} }$-lattice of rank 2 inside of ${\displaystyle \mathbb {C} }$[1] ^{pg 158}. Conversely, given an integral lattice ${\displaystyle \Lambda }$ of rank ${\displaystyle 2}$ inside of ${\displaystyle \mathbb {C} }$, there is an embedding of the complex torus ${\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda }$ into ${\displaystyle \mathbb {P} ^{2}}$ from the Weierstrass P function[1] ^{pg 165}. This isomorphic correspondence ${\displaystyle \phi$ :\mathbb {C} /\Lambda \to E(\mathbb {C} )} is given by

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

and holds up to homothety of the lattice ${\displaystyle \Lambda }$, which is the equivalence relation

${\displaystyle z\Lambda \sim \Lambda }$ for ${\displaystyle z\in \mathbb {C} -\{0\}}$

It is standard to then write the lattice in the form ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau }$ for ${\displaystyle \tau \in {\mathfrak {h}}}$, an element of the upper half-plane, since the lattice ${\displaystyle \Lambda }$ could be multiplied by ${\displaystyle \omega _{1}^{-1}}$, and ${\displaystyle \tau ,-\tau }$ both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over ${\displaystyle \mathbb {C} }$. There is an additional equivalence of curves given by the action of the

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

where an elliptic curve defined by the lattice ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau }$ is isomorphic to curves defined by the lattice ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau '}$ given by the modular action

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

Then, the moduli stack of elliptic curves over ${\displaystyle \mathbb {C} }$ is given by the stack quotient

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

Note some authors construct this moduli space by instead using the action of the Modular group ${\displaystyle {\text{PSL}}_{2}(\mathbb {Z} )={\text{SL}}_{2}(\mathbb {Z} )/\{\pm I\}}$. In this case, the points in ${\displaystyle {\mathcal {M}}_{1,1}}$ having only trivial stabilizers are dense.

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

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

and a solution ${\displaystyle (t,s)}$, generically for j-invariant ${\displaystyle j\neq 0,1728}$, there is the ${\displaystyle \mathbb {Z} /2}$-involution sending ${\displaystyle (t,s)\mapsto (t,-s)}$. In the special case of a curve with complex multiplication

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

there the ${\displaystyle \mathbb {Z} /4}$-involution sending ${\displaystyle (t,s)\mapsto (-t,{\sqrt {-1}}\cdot s)}$. The other special case is when ${\displaystyle a=0}$, so a curve of the form

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

there is the ${\displaystyle \mathbb {Z} /6}$-involution sending ${\displaystyle (t,s)\mapsto (\zeta _{3}s,-t)}$ where ${\displaystyle \zeta _{3}}$ is the third root of unity ${\displaystyle e^{2\pi i/3}}$.

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

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

It is useful to consider this space because it helps visualize the stack ${\displaystyle {\mathcal {M}}_{1,1}}$. From the quotient map

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

the image of ${\displaystyle D}$ is surjective and its interior is injective[4]^{pg 78}. Also, the points on the boundary can be identified with their mirror image under the involution sending ${\displaystyle {\text{Re}}(z)\mapsto -{\text{Re}}(z)}$, so ${\displaystyle {\mathcal {M}}_{1,1}}$ can be visualized as the projective curve ${\displaystyle \mathbb {P} ^{1}}$ with a point removed at infinity[5]^{pg 52}.

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

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

The degree ${\displaystyle k}$ action is given by

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

hence the trivial line bundle ${\displaystyle \mathbb {C} \times {\mathfrak {h}}\to {\mathfrak {h}}}$ with the degree ${\displaystyle k}$ action descends to a unique line bundle denoted ${\displaystyle {\mathcal {L}}^{\otimes k}}$. Notice the action on the factor ${\displaystyle \mathbb {C} }$ is a representation of ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$ on ${\displaystyle \mathbb {Z} }$ hence such representations can be tensored together, showing ${\displaystyle {\mathcal {L}}^{\otimes k}\otimes {\mathcal {L}}^{\otimes l}\cong {\mathcal {L}}^{\otimes (k+l)}}$. The sections of ${\displaystyle {\mathcal {L}}^{\otimes k}}$ are then functions sections ${\displaystyle f\in \Gamma (\mathbb {C} \times {\mathfrak {h}})}$ compatible with the action of ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$, or equivalently, functions ${\displaystyle f:{\mathfrak {h}}\to \mathbb {C} }$ such that

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

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

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

this is because in order to compactify the stack ${\displaystyle {\mathcal {M}}_{1,1}}$, a point at infinity must be added, which is done through a gluing process by gluing the ${\displaystyle q}$-disk (where a modular function has its ${\displaystyle q}$-expansion)[2]^{pgs 29-33}.

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

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

Because the ${\displaystyle \tau }$ in ${\displaystyle {\mathfrak {h}}}$ can vary in this action, there is an induced ${\displaystyle \mathbb {Z} ^{2}}$-action on ${\displaystyle \mathbb {C} \times {\mathfrak {h}}}$

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

giving the quotient space

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

by projecting onto ${\displaystyle {\mathfrak {h}}}$.

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

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

which is matrix multiplication on the right, so

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

See also

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

References

1. Silverman, Joseph H., 1955- (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184.CS1 maint: multiple names: authors list (link)
2. Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math].
3. Galbraith, Steven. "Elliptic Curves" (PDF). Archived from the original on |archive-url= requires |archive-date= (help).
4. Serre, Jean-Pierre. (1973). A Course in Arithmetic. New York, NY: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550.
5. "3: The Moduli stack of elliptic curves". Topological modular forms (PDF). Douglas, Christopher L.,, Francis, John, 1982-, Henriques, André G. (André Gil), 1977-, Hill, Michael A. (Michael Anthony),. Providence, Rhode Island. ISBN 978-1-4704-1884-7. OCLC 884782304. Archived from the original (PDF) on 9 Jun 2020.CS1 maint: extra punctuation (link) CS1 maint: others (link)

• Hain, Richard (2008), Lectures on Moduli Spaces of Elliptic Curves, arXiv:0812.1803, Bibcode:2008arXiv0812.1803H
• Lurie, Jacob (2009), A survey of elliptic cohomology (PDF)
• Olsson, Martin (2016), Algebraic spaces and stacks, Colloquium Publications, 62, American Mathematical Society, ISBN 978-1470427986

External links

• moduli+stack+of+elliptic+curves in nLab
• "The moduli stack of elliptic curves", Stacks project