Moduli stack of elliptic curves

Short description: Algebraic stack in mathematics

In mathematics, the moduli stack of elliptic curves, denoted as ℳ1,1 or ℳell, is an algebraic stack over Spec(ℤ) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves ℳg,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 ℳ1,1.

## Contents

* 1 Properties
* 1.1 Smooth Deligne-Mumford stack
* 1.2 j-invariant
* 2 Construction over the complex numbers
* 2.1 Stacky/Orbifold points
* 2.1.1 Representing involutions of plane curves
* 2.2 Fundamental domain and visualization
* 2.3 Line bundles and modular functions
* 2.3.1 Modular forms
* 2.4 Universal curves
* 2.4.1 Versal curve
* 2.4.2 SL2-action on Z2
* 3 See also
* 4 References
* 5 External links

## Properties

### Smooth Deligne-Mumford stack

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

### j-invariant

There is a proper morphism of ℳ1,1 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 ℂ is classified by its periods. Given a basis for its integral homology α,β∈H1(E,ℤ) and a global holomorphic differential form ω∈Γ(E,ΩE1) (which exists since it is smooth and the dimension of the space of such differentials is equal to the genus, 1), the integrals[∫αω∫βω]=[ω1ω2]give the generators for a ℤ-lattice of rank 2 inside of ℂ[1] pg 158. Conversely, given an integral lattice Λ of rank 2 inside of ℂ, there is an embedding of the complex torus EΛ=ℂ/Λ into ℙ2 from the Weierstrass P function[1] pg 165. This isomorphic correspondence ϕ:ℂ/Λ->E(ℂ) is given byz↦[℘(z,Λ),℘′(z,Λ),1]∈ℙ2(ℂ)and holds up to homothety of the lattice Λ, which is the equivalence relationzΛ∼Λforz∈ℂ∖{0}It is standard to then write the lattice in the form ℤ⊕ℤ⋅τ for τ∈𝔥, an element of the upper half-plane, since the lattice Λ could be multiplied by ω1−1, and τ,−τ both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over ℂ. There is an additional equivalence of curves given by the action of theSL2(ℤ)={(abcd)∈Mat2,2(ℤ):ad−bc=1}where an elliptic curve defined by the lattice ℤ⊕ℤ⋅τ is isomorphic to curves defined by the lattice ℤ⊕ℤ⋅τ′ given by the modular action(abcd)⋅τ=aτ+bcτ+d=τ′Then, the moduli stack of elliptic curves over ℂ is given by the stack quotientℳ1,1≅[SL2(ℤ)∖𝔥]Note some authors construct this moduli space by instead using the action of the Modular group PSL2(ℤ)=SL2(ℤ)/{±I}. In this case, the points in ℳ1,1 having only trivial stabilizers are dense.

Fundamental domains of the action of SL2(ℤ) on the upper half-plane are shown here as pairs of ideal triangles of different colors sharing an edge. The "standard" fundamental domain is shown with darker edges. Suitably identifying points on the boundary of this region, we obtain the coarse moduli space of elliptic curves. The stacky points at τ=i and τ=e2πi/3,eπi/3 are on the boundary of this region.

### Stacky/Orbifold points

Generically, the points in ℳ1,1 are isomorphic to the classifying stack B(ℤ/2) since every elliptic curve corresponds to a double cover of ℙ1, so the 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 τ=i, and the points corresponding to the stabilizer of order 6 correspond to τ=e2πi/3,eπi/3[4]pg 78.

#### Representing involutions of plane curves

Given a plane curve by its Weierstrass equationy2=x3+ax+band a solution (t,s), generically for j-invariant j≠0,1728, there is the ℤ/2-involution sending (t,s)↦(t,−s). In the special case of a curve with complex multiplicationy2=x3+axthere the ℤ/4-involution sending (t,s)↦(−t,−1⋅s). The other special case is when a=0, so a curve of the formy2=x3+b there is the ℤ/6-involution sending (t,s)↦(ζ3t,−s) where ζ3 is the third root of unity e2πi/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 subsetD={z∈𝔥:|z|≥1 and Re(z)≤1/2}It is useful to consider this space because it helps visualize the stack ℳ1,1. From the quotient map𝔥->SL2(ℤ)∖𝔥the 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 Re(z)↦−Re(z), so ℳ1,1 can be visualized as the projective curve ℙ1 with a point removed at infinity[5]pg 52.

### Line bundles and modular functions

There are line bundles ℒ⊗k over the moduli stack ℳ1,1 whose sections correspond to modular functions f on the upper-half plane 𝔥. On ℂ×𝔥 there are SL2(ℤ)-actions compatible with the action on 𝔥 given bySL2(ℤ)×ℂ×𝔥->ℂ×𝔥The degree k action is given by(abcd):(z,τ)↦((cτ+d)kz,aτ+bcτ+d)hence the trivial line bundle ℂ×𝔥->𝔥 with the degree k action descends to a unique line bundle denoted ℒ⊗k. Notice the action on the factor ℂ is a representation of SL2(ℤ) on ℤ hence such representations can be tensored together, showing ℒ⊗k⊗ℒ⊗l≅ℒ⊗(k+l). The sections of ℒ⊗k are then functions sections f∈Γ(ℂ×𝔥) compatible with the action of SL2(ℤ), or equivalently, functions f:𝔥->ℂ such thatf((abcd)⋅τ)=(cτ+d)kf(τ) 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 compactificationℒ⊗k‾->ℳ‾1,1this is because in order to compactify the stack ℳ1,1, 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 ℰ->ℳ1,1 is a two step process: (1) construct a versal curve ℰ𝔥->𝔥 and then (2) show this behaves well with respect to the SL2(ℤ)-action on 𝔥. Combining these two actions together yields the quotient stack[(SL2(ℤ)⋉ℤ2)∖ℂ×𝔥]

#### Versal curve

Every rank 2 ℤ-lattice in ℂ induces a canonical ℤ2-action on ℂ. As before, since every lattice is homothetic to a lattice of the form (1,τ) then the action (m,n) sends a point z∈ℂ to(m,n)⋅z↦z+m⋅1+n⋅τBecause the τ in 𝔥 can vary in this action, there is an induced ℤ2-action on ℂ×𝔥(m,n)⋅(z,τ)↦(z+m⋅1+n⋅τ,τ)giving the quotient spaceℰ𝔥->𝔥by projecting onto 𝔥.

#### SL2-action on Z2

There is a SL2(ℤ)-action on ℤ2 which is compatible with the action on 𝔥, meaning given a point z∈𝔥 and a g∈SL2(ℤ), the new lattice g⋅z and an induced action from ℤ2⋅g, which behaves as expected. This action is given by(abcd):(m,n)↦(m,n)⋅(abcd)which is matrix multiplication on the right, so(m,n)⋅(abcd)=(am+cn,bm+dn)

## See also

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

## References

1. ↑ 1.0 1.1 Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184.
2. ↑ 2.0 2.1 Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
3. ↑ Galbraith, Steven. "Elliptic Curves". Mathematics of Public Key Cryptography. Cambridge University Press. https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf.
4. ↑ 4.0 4.1 Serre, Jean-Pierre (1973). A Course in Arithmetic. New York: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550.
5. ↑ Henriques, André G. "The Moduli stack of elliptic curves". in Douglas, Christopher L.. Topological modular forms. Providence, Rhode Island. ISBN 978-1-4704-1884-7. OCLC 884782304. https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf.

* Hain, Richard (2008), Lectures on Moduli Spaces of Elliptic Curves, Bibcode: 2008arXiv0812.1803H
* Lurie, Jacob (2009), A survey of elliptic cohomology, http://www.math.harvard.edu/~lurie/papers/survey.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, http://stacks.math.columbia.edu/tag/072K

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