Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as ${\mathcal {M}}_{1,1}$ or ${\mathcal {M}}_{ell}$ , 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 $\mathbb {C}$ [1] ^{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[1] ^{pg 165}. This isomorphic correspondence ${\displaystyle \phi$ is given by

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

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

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

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.

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[2] ^{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[3] ^{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 $\tau =e^{2\pi i/3},e^{\pi i/3}$ [4]^{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 $(t,s)\mapsto (\zeta _{3}s,-t)$ 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[4]^{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[5]^{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

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

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)[2]^{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 ${\mathfrak {h}}$ .

SL₂-action on Z²

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

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)
Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math].
Galbraith, Steven. "Elliptic Curves" (PDF). Archived from the original on |archive-url= requires |archive-date= (help).
Serre, Jean-Pierre. (1973). A Course in Arithmetic. New York, NY: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550.
"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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[:0-1] 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)

[:2-2] 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).

[:1-4] 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)