Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M e l l {\displaystyle {\mathcal {M}}_{\mathrm {ell} }}, is an algebraic stack over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb {Z} )} classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves M g , n {\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 S {\displaystyle S} to it correspond to elliptic curves over S {\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 M 1 , 1 {\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 Spec ( Z ) {\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 M 1 , 1 {\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 C {\displaystyle \mathbb {C} } is classified by its periods. Given a basis for its integral homology α , β ∈ H 1 ( E , Z ) {\displaystyle \alpha ,\beta \in H_{1}(E,\mathbb {Z} )} and a global holomorphic differential form ω ∈ Γ ( E , Ω E 1 ) {\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[ ∫ α ω ∫ β ω ] = [ ω 1 ω 2 ] {\displaystyle {\begin{bmatrix}\int _{\alpha }\omega &\int _{\beta }\omega \end{bmatrix}}={\begin{bmatrix}\omega _{1}&\omega _{2}\end{bmatrix}}}give the generators for a Z {\displaystyle \mathbb {Z} }-lattice of rank 2 inside of C {\displaystyle \mathbb {C} } pg 158. Conversely, given an integral lattice Λ {\displaystyle \Lambda } of rank 2 {\displaystyle 2} inside of C {\displaystyle \mathbb {C} }, there is an embedding of the complex torus E Λ = C / Λ {\displaystyle E_{\Lambda }=\mathbb {C} /\Lambda } into P 2 {\displaystyle \mathbb {P} ^{2}} from the Weierstrass P function pg 165. This isomorphic correspondence ϕ : C / Λ → E ( C ) {\displaystyle \phi :\mathbb {C} /\Lambda \to E(\mathbb {C} )} is given byz ↦ [ ℘ ( z , Λ ) , ℘ ′ ( z , Λ ) , 1 ] ∈ P 2 ( C ) {\displaystyle z\mapsto [\wp (z,\Lambda ),\wp '(z,\Lambda ),1]\in \mathbb {P} ^{2}(\mathbb {C} )}and holds up to homothety of the lattice Λ {\displaystyle \Lambda }, which is the equivalence relationz Λ ∼ Λ for z ∈ C ∖ { 0 } {\displaystyle z\Lambda \sim \Lambda ~{\text{for}}~z\in \mathbb {C} \setminus \{0\}}It is standard to then write the lattice in the form Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } for τ ∈ h {\displaystyle \tau \in {\mathfrak {h}}}, an element of the upper half-plane, since the lattice Λ {\displaystyle \Lambda } could be multiplied by ω 1 − 1 {\displaystyle \omega _{1}^{-1}}, and τ , − τ {\displaystyle \tau ,-\tau } both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over C {\displaystyle \mathbb {C} }. There is an additional equivalence of curves given by the action of theSL 2 ( Z ) = { ( a b c d ) ∈ Mat 2 , 2 ( Z ) : a d − b c = 1 } {\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 Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } is isomorphic to curves defined by the lattice Z ⊕ Z ⋅ τ ′ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau '} given by the modular action( a b c d ) ⋅ τ = a τ + b c τ + d = τ ′ {\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 C {\displaystyle \mathbb {C} } is given by the stack quotientM 1 , 1 ≅ [ SL 2 ( Z ) ∖ h ] {\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 PSL 2 ( Z ) = SL 2 ( Z ) / { ± I } {\displaystyle {\text{PSL}}_{2}(\mathbb {Z} )={\text{SL}}_{2}(\mathbb {Z} )/\{\pm I\}}. In this case, the points in M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} having only trivial stabilizers are dense.

{\displaystyle \qquad }

Stacky/Orbifold points

Generically, the points in M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} are isomorphic to the classifying stack B ( Z / 2 ) {\displaystyle B(\mathbb {Z} /2)} since every elliptic curve corresponds to a double cover of P 1 {\displaystyle \mathbb {P} ^{1}}, so the Z / 2 {\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 pg 10-11 corresponding to elliptic curves with j {\displaystyle j}-invariant equal to 1728 {\displaystyle 1728} and 0 {\displaystyle 0} where the automorphism groups are of order 4, 6, respectively pg 170. One point in the Fundamental domain with stabilizer of order 4 {\displaystyle 4} corresponds to τ = i {\displaystyle \tau =i}, and the points corresponding to the stabilizer of order 6 {\displaystyle 6} correspond to τ = e 2 π i / 3 , e π i / 3 {\displaystyle \tau =e^{2\pi i/3},e^{\pi i/3}}pg 78.

Representing involutions of plane curves

Given a plane curve by its Weierstrass equationy 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b}and a solution ( t , s ) {\displaystyle (t,s)}, generically for j-invariant j ≠ 0 , 1728 {\displaystyle j\neq 0,1728}, there is the Z / 2 {\displaystyle \mathbb {Z} /2}-involution sending ( t , s ) ↦ ( t , − s ) {\displaystyle (t,s)\mapsto (t,-s)}. In the special case of a curve with complex multiplicationy 2 = x 3 + a x {\displaystyle y^{2}=x^{3}+ax}there the Z / 4 {\displaystyle \mathbb {Z} /4}-involution sending ( t , s ) ↦ ( − t , − 1 ⋅ s ) {\displaystyle (t,s)\mapsto (-t,{\sqrt {-1}}\cdot s)}. The other special case is when a = 0 {\displaystyle a=0}, so a curve of the formy 2 = x 3 + b {\displaystyle y^{2}=x^{3}+b} there is the Z / 6 {\displaystyle \mathbb {Z} /6}-involution sending ( t , s ) ↦ ( ζ 3 t , − s ) {\displaystyle (t,s)\mapsto (\zeta _{3}t,-s)} where ζ 3 {\displaystyle \zeta _{3}} is the third root of unity e 2 π i / 3 {\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 subsetD = { z ∈ h : | z | ≥ 1 and Re ( z ) ≤ 1 / 2 } {\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 M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}}. From the quotient maph → SL 2 ( Z ) ∖ h {\displaystyle {\mathfrak {h}}\to {\text{SL}}_{2}(\mathbb {Z} )\backslash {\mathfrak {h}}}the image of D {\displaystyle D} is surjective and its interior is injectivepg 78. Also, the points on the boundary can be identified with their mirror image under the involution sending Re ( z ) ↦ − Re ( z ) {\displaystyle {\text{Re}}(z)\mapsto -{\text{Re}}(z)}, so M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} can be visualized as the projective curve P 1 {\displaystyle \mathbb {P} ^{1}} with a point removed at infinitypg 52.

Line bundles and modular functions

There are line bundles L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}} over the moduli stack M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} whose sections correspond to modular functions f {\displaystyle f} on the upper-half plane h {\displaystyle {\mathfrak {h}}}. On C × h {\displaystyle \mathbb {C} \times {\mathfrak {h}}} there are SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}-actions compatible with the action on h {\displaystyle {\mathfrak {h}}} given bySL 2 ( Z ) × C × h → C × h {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )\times {\displaystyle \mathbb {C} \times {\mathfrak {h}}}\to {\displaystyle \mathbb {C} \times {\mathfrak {h}}}}The degree k {\displaystyle k} action is given by( a b c d ) : ( z , τ ) ↦ ( ( c τ + d ) k z , a τ + b c τ + d ) {\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 C × h → h {\displaystyle \mathbb {C} \times {\mathfrak {h}}\to {\mathfrak {h}}} with the degree k {\displaystyle k} action descends to a unique line bundle denoted L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}}. Notice the action on the factor C {\displaystyle \mathbb {C} } is a representation of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} on Z {\displaystyle \mathbb {Z} } hence such representations can be tensored together, showing L ⊗ k ⊗ L ⊗ l ≅ L ⊗ ( k + l ) {\displaystyle {\mathcal {L}}^{\otimes k}\otimes {\mathcal {L}}^{\otimes l}\cong {\mathcal {L}}^{\otimes (k+l)}}. The sections of L ⊗ k {\displaystyle {\mathcal {L}}^{\otimes k}} are then functions sections f ∈ Γ ( C × h ) {\displaystyle f\in \Gamma (\mathbb {C} \times {\mathfrak {h}})} compatible with the action of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}, or equivalently, functions f : h → C {\displaystyle f:{\mathfrak {h}}\to \mathbb {C} } such thatf ( ( a b c d ) ⋅ τ ) = ( c τ + d ) k f ( τ ) {\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 compactificationL ⊗ k ¯ → M ¯ 1 , 1 {\displaystyle {\overline {{\mathcal {L}}^{\otimes k}}}\to {\overline {\mathcal {M}}}_{1,1}}this is because in order to compactify the stack M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}}, a point at infinity must be added, which is done through a gluing process by gluing the q {\displaystyle q}-disk (where a modular function has its q {\displaystyle q}-expansion)pgs 29-33.

Universal curves

Constructing the universal curves E → M 1 , 1 {\displaystyle {\mathcal {E}}\to {\mathcal {M}}_{1,1}} is a two step process: (1) construct a versal curve E h → h {\displaystyle {\mathcal {E}}_{\mathfrak {h}}\to {\mathfrak {h}}} and then (2) show this behaves well with respect to the SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}-action on h {\displaystyle {\mathfrak {h}}}. Combining these two actions together yields the quotient stack[ ( SL 2 ( Z ) ⋉ Z 2 ) ∖ C × h ] {\displaystyle [({\text{SL}}_{2}(\mathbb {Z} )\ltimes \mathbb {Z} ^{2})\backslash \mathbb {C} \times {\mathfrak {h}}]}

Versal curve

Every rank 2 Z {\displaystyle \mathbb {Z} }-lattice in C {\displaystyle \mathbb {C} } induces a canonical Z 2 {\displaystyle \mathbb {Z} ^{2}}-action on C {\displaystyle \mathbb {C} }. As before, since every lattice is homothetic to a lattice of the form ( 1 , τ ) {\displaystyle (1,\tau )} then the action ( m , n ) {\displaystyle (m,n)} sends a point z ∈ C {\displaystyle z\in \mathbb {C} } to( m , n ) ⋅ z ↦ z + m ⋅ 1 + n ⋅ τ {\displaystyle (m,n)\cdot z\mapsto z+m\cdot 1+n\cdot \tau }Because the τ {\displaystyle \tau } in h {\displaystyle {\mathfrak {h}}} can vary in this action, there is an induced Z 2 {\displaystyle \mathbb {Z} ^{2}}-action on C × h {\displaystyle \mathbb {C} \times {\mathfrak {h}}}( m , n ) ⋅ ( z , τ ) ↦ ( z + m ⋅ 1 + n ⋅ τ , τ ) {\displaystyle (m,n)\cdot (z,\tau )\mapsto (z+m\cdot 1+n\cdot \tau ,\tau )}giving the quotient spaceE h → h {\displaystyle {\mathcal {E}}_{\mathfrak {h}}\to {\mathfrak {h}}}by projecting onto h {\displaystyle {\mathfrak {h}}}.

SL 2 -action on Z 2

There is a SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}-action on Z 2 {\displaystyle \mathbb {Z} ^{2}} which is compatible with the action on h {\displaystyle {\mathfrak {h}}}, meaning given a point z ∈ h {\displaystyle z\in {\mathfrak {h}}} and a g ∈ SL 2 ( Z ) {\displaystyle g\in {\text{SL}}_{2}(\mathbb {Z} )}, the new lattice g ⋅ z {\displaystyle g\cdot z} and an induced action from Z 2 ⋅ g {\displaystyle \mathbb {Z} ^{2}\cdot g}, which behaves as expected. This action is given by( a b c d ) : ( m , n ) ↦ ( m , n ) ⋅ ( a b c d ) {\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( m , n ) ⋅ ( a b c d ) = ( a m + c n , b m + d n ) {\displaystyle (m,n)\cdot {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=(am+cn,bm+dn)}

External links

at the nLab
, Stacks project