moduli stack of elliptic curves



A moduli stack of elliptic curves is a moduli stack for elliptic curves, hence a stack ell\mathcal{M}_{ell} such that for XX any other suitable space, the groupoid of maps X ellX \to \mathcal{M}_{ell} and homotopies between them is equivalent to that of XX-parameterized elliptic curves with equivalences between these. (There are some variants of ell\mathcal{M}_{ell} corresponding to the choice of which singularities and degeneracies of elliptic curves are taken into account.)

This is formalized in algebraic geometry, hence XX here is a scheme over the integers in general. The moduli stack always has a tautological construction as a “sheaf of groupoids” (a stack, whence the name) over the site of affine schemes, given by sending any Spec(R)Spec(R) to the groupoid of suitable elliptic curves over Spec(R)Spec(R). For concrete computations it typically helps to know that the moduli stack of elliptic curbes is represented by a geometric stack, dually given by a Hopf algebroid.

The moduli stack ell\mathcal{M}_{ell} has a compactification ell¯\mathcal{M}_{\overline{ell}} obtained by adding the nodal cubic curve, and often (but not always) this compactified version is the default meaning of “moduli stack of elliptic curves”. Adding also the cuspidal cubic curve and hence all cubic curves produces the full moduli stack cub\mathcal{M}_{cub} of cubic curves, inside which ell\mathcal{M}_{ell} sits as the locus of non-singular curves.

Since an elliptic curve is a genus-1 algebraic curve with a marked point (the neutral element of the group structure), ell\mathcal{M}_{ell} is equivalently the moduli stack of algebraic curves for genus g=1g = 1 with n=1n = 1 punctures, and as such is often equivalently written

1,1= ell. \mathcal{M}_{1,1} = \mathcal{M}_{ell} \,.

A special class of cases which is much simpler than the general case but still of paramount interest is the moduli stack of elliptic curves over the complex numbers, hence of maps Spec(𝒞) ellSpec(\mathcal{C}) \to \mathcal{M}_{ell}. These are just complex tori/Riemannian manifolds of genus 1 which may be identified with quotients of the complex plane by a framed lattice well-defined up to Möbius transformations, and so in this case the moduli stack of elliptic curves is just the homotopy quotient (the orbifold quotient) of the upper half plane by the action of the modular group. This case is considered below in

Below that is the

as an algebraic stack ell\mathcal{M}_{ell}. This is still not the most refined description: by the Goerss-Hopkins-Miller theorem the assignment to an elliptic curve of its elliptic spectrum lifts the ordinary structure sheaf 𝒪\mathcal{O} of ell\mathcal{M}_{ell} to a higher structure sheaf 𝒪 top\mathcal{O}^{top} of E-∞ rings in a way that makes ( ell,𝒪 op)(\mathcal{M}_{ell}, \mathcal{O}^{op}) a spectral Deligne-Mumford stack. The global sections of this structure sheaf yield the spectrum tmf of topological modular forms:

See also A Survey of Elliptic Cohomology - elliptic curves for more.

Description over the complex numbers

An elliptic curve ESpec()E \to Spec(\mathbb{C}) over the complex numbers is determined, up to non-canonical isomorphism, by its j-invariant

j(E). j(E) \in \mathbb{C} \,.

Here every complex number appears as a value, and therefore the moduli space of elliptic cuves a priori is not compact.

A compactification of the moduli space is obtained by including also elliptic curves with nodal singularity.

Upper half plane

The upper half plane 𝔥\mathfrak{h} is in bijection with framed lattices in the complex plane \mathbb{C}, which in turn is in bijection with isomorphism classes of framed elliptic curves over \mathbb{C}

𝔥{framedlatticesin}{framedellipticcurvesover}/ \mathfrak{h} \simeq \{framed\;lattices\;in\;\mathbb{C}\} \simeq \{framed\;elliptic\;curves\;over\;\mathbb{C}\}/_\sim

and we have

{ellipticcurvesover} 𝔥/SL 2() \{elliptic\;curves\;over\;\mathbb{C}\}_\sim \simeq \mathfrak{h}/{SL_2(\mathbb{Z})}

where the special linear group over the integers

SL 2()={(a b c d)adcd=1} SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}

acts as the modular group by Möbius transformations

τaτ+bcτ+d. \tau \mapsto \frac{a \tau + b}{c \tau + d} \,.

The naive moduli space and its problems



M 1,1:=𝔥/SL 2() M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})

for the plain quotient of the upper half plane by the above group action.


A homolorphic family of elliptic curves over a complex manifold TT is

  • a holomorphic function π:XT\pi : X \to T

  • together with a section s:TXs : T \to X of π\pi such that for any tTt \in T the pair (X t,s(t))(X_t, s(t)) is an elliptic curve (using the first definition above).

For every family

X π T \array{ X \\ \downarrow^{\mathrlap{\pi}} \\ T }

we would like to have FM 1,1F \to M_{1,1} such that there is a pullback

Xϕ *F F T ϕ M 1,1 \array{ X \simeq \phi^* F &\longrightarrow& F \\ \downarrow & & \downarrow \\ T &\stackrel{\phi}{\longrightarrow}& M_{1,1} }


ϕ:t[X t,s(t)] \phi: t \mapsto [X_t, s(t)]

such that

  • ϕ:TM 1,1\phi : T \to M_{1,1} is a holomorphic map

  • every holomorphic map TM 1,1T \to M_{1,1} corresponds to a family over tt;

  • there is a universal family over M 1,1M_{1,1}

This is impossible . One can construct explicit counterexamples. These counterexamples involve elliptic curves with nontrivial automorphisms.

For instance

{(x,y,z) 2×X:y 2=x(x1)(xλ)}X:= 1{0,1,} \{ (x,y,z) \in \mathbb{P}^2 \times X : y^2 = x(x-1)(x-\lambda) \} \to X := \mathbb{P}^1 - \{0,1,\infty\}

but see the discussion at moduli space for a discussion of the statement “it’s the automorphisms that prevent the moduli space from existing”

Moduli space of framed elliptic curves


2×𝔥 \mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}

given by

(m,n):(z,τ)(z+mτ+n,τ) (m,n) : (z,\tau) \mapsto (z + m \tau + n, \tau)

Then consider the family

E:=/ 2×𝔥 𝔥 \array{ E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h} \\ \downarrow \\ \mathfrak{h} }

is a family of elliptic curves over 𝔥\mathfrak{h}

and E τ=/Λ τE_\tau = \mathbb{C}/{\Lambda_\tau} with

Λ τ:=1τ \Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau

is a family of framed elliptic curves.


The space 𝔥\mathfrak{h} with the family E𝔥E \to \mathfrak{h} is a fine moduli space for framed elliptic curves.

Consider any map ϕ:T𝔥\phi : T \to \mathfrak{h}

with pullback of the universal family

X?ϕ *E E T ϕ 𝔥 \array{ X \stackrel{?}{\to} \phi^* E &\to & E \\ \downarrow && \downarrow \\ T &\stackrel{\phi}{\to}& \mathfrak{h} }

claim for every point tTt \in T there is an open neighbourhood t 0UTt_0 \in U \hookrightarrow T such that one can choose 1-forms ω t\omega_t on X τX_\tau which vary holomorphically with respect to tt.

Notice that locally every family of elliptic curves is framed (since we can locally extend a choice of basis for H 1H_1). So

𝔥 SL 2() M 1,1 Id M 1,1 \array{ && \mathfrak{h} \\ && \downarrow^{SL_2(\mathbb{Z})} \\ M_{1,1} &\stackrel{Id}{\to}& M_{1,1} }

at ii and ρ=e 2πi/6\rho = e^{2\pi i/6} , C={±I}C = \{\pm I\}

isn’t locally liftable at ii and ρ\rho so it is not a univresal family of unframed curves.

Moduli stack/orbifold of elliptic curves


Consider the global quotient stack orbifold

1,1:=𝔥//SL 2() \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})

of the upper half plane by the action of the special linear group over the integers.

This is the moduli stack of elliptic curves.

Compactified moduli stack

Consider the complex analytic parameterization over the annulus

{q0<q<1} \{q \in \mathbb{C} | 0 \lt {\vert q \vert} \lt 1 \}

of elliptic curves

E q/q . E_q \coloneqq \mathbb{C}/q^{\mathbb{Z}} \,.

This has an extension to the origin, where E 0E_0 is a nodal curve. Algebraically, in a formal neighbourhood of the origin, hence over Spec([[q]])Spec(\mathbb{Z}[ [q] ]), this is the Tate curve.

e.g. (Lurie, section 4.3).

Description over general schemes

For SS a scheme, a cubic curve over SS is a scheme p:XSp \colon X \to S over SS equipped with a section e:SXe \colon S \to X and such that Zariski locally on SS, XX is given by an equation in S 2\mathbb{P}_S^2 of the form

y 2+a 1xy=x 3+a 2x 2+a 4x+a 6 y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6

such that e:SXe \colon S \to X is the line at infinity.

Euivalently this says that pp is a proper flat morphism with a section contained in the smooth locus whose fibers are geometrically integral curves of arithmetic genus one.

Write cub\mathcal{M}_{cub} for the moduli stack of such cubic curves. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant ΔH 0( cub,ω 12)\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})

ell cub FG. \mathcal{M}_{ell} \hookrightarrow \mathcal{M}_{cub} \longrightarrow \mathcal{M}_{FG}.

See at elliptic curve for details. (A textbook account is in Silverman 09, III, a review with an eye towards tmf is in Mathew, section 3).

Two standard versions of Hopf algebroids representing ell¯\mathcal{M}_{\overline{ell}} as a geometric stack are usefully reviewed in (Mathew, section 4).

As a derived scheme in E E_\infty-geometry

By the Goerss-Hopkins-Miller theorem the structure sheaf 𝒪\mathcal{O} of the moduli stack of elliptic curves lifts to a sheaf 𝒪 top\mathcal{O}^{top} of E-∞ rings which over a given elliptic curve is the corresponding elliptic spectrum.

By (Lurie (Survey), theorem 4.1), this yields a spectral Deligne-Mumford stack refinement

ell der( ell,𝒪 top) \mathcal{M}_{ell}^{der} \coloneqq (\mathcal{M}_{ell}, \mathcal{O}^{top})

which is the moduli stack of derived elliptic curves, in that there is a natural equivalence in E-∞ rings AA of the form

Hom(Spec(A), ell der)E(A), Hom(Spec(A), \mathcal{M}_{ell}^{der}) \simeq E(A) \,,

where on the left we have maps of structured (∞,1)-toposes and on the right the ∞-groupoid of derived elliptic curves over AA.

This is based on the representability theorem (Lurie (Survey), prop. 4.1, Lurie (Representability)).

In this derived picture the compactified dericed moduli space is obtained by gluing in the spectrum of Tate K-theory KO[[q]]KU[[q]]/ 2KO[ [q] ] \simeq KU[ [q] ]/\mathbb{Z}_2 by forming the homotopy pushout

Spec(K((q))) ell der Spec(K[[q]]) ell¯ der. \array{ Spec(K((q))) &\longrightarrow& \mathcal{M}_{ell}^{der} \\ \downarrow && \downarrow \\ Spec(K[ [q] ]) &\longrightarrow& \mathcal{M}_{\overline{ell}}^{der} } \,.

(Lurie(Survey), p. 33). Again, the underlying ordinary Deligne-Mumford stack is the ordinary ¯\mathcal{M}_{\overline{\ell}}.


Finite covers

The moduli space ell[n]\mathcal{M}_{ell}[n] of elliptic curves with level-n structure (for some nn \in \mathbb{N}) provides a finite covering of ell\mathcal{M}_{ell} (similarly for the compactifications).

(Over the complex numbers this is the modular curve).


H 1( 1,1,)=/12 H_1(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12\mathbb{Z}
H 1( 1,1,)=0 H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0
H 2( 1,1,)=/12 H^2(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12 \mathbb{Z}
H ( 1,1,)H (M 1,1,) H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q}) \simeq H_\bullet(M_{1,1}, \mathbb{Q})

and similarly for integral cohomology

χ( 1,1)=112 \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
Pic( 1,1)/12 Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}

Euler characteristic

The orbifold Euler characteristic of the moduli space of complex elliptic curves is given by the Riemann zeta function

χ( 1,1)=ζ(1)=112. \chi(\mathcal{M}_{1,1}) = \zeta(-1) = - \frac{1}{12} \,.

This is a special case of the result in (Zagier-Harer 86) discussed at moduli space of curves. See also the first page here: pdf.

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering moduli spaceof level-n structures (modular curve)
*=Spec()\ast = Spec(\mathbb{Z})\toSpec([[q]])Spec(\mathbb{Z}[ [q] ])\to ell¯[n]\mathcal{M}_{\overline{ell}}[n]
structure group /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} SL 2(/n)\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})} (modular group)
1dTori\mathcal{M}_{1dTori}\hookrightarrow Tate\mathcal{M}_{Tate}\hookrightarrow ell¯\mathcal{M}_{\overline{ell}}\hookrightarrow cub\mathcal{M}_{cub}\to FG\mathcal{M}_{FG}
moduli stackof 1d toriof Tate curvesof elliptic curvesof cubic curvesof formal groups
𝒪 Σ top\mathcal{O}^{top}_{\Sigma}KUKU[[q]]KU[ [q] ]elliptic spectrumcomplex oriented cohomology theory
Γ(,𝒪 top)=\Gamma(-, \mathcal{O}^{top}) = (KO \hookrightarrow KU) = KR-theoryTate K-theory (KO[[q]]KU[[q]]KO[ [q] ] \hookrightarrow KU[ [q] ])(Tmf \to Tmf(n)) (modular equivariant elliptic cohomology)tmf𝕊\mathbb{S}


Introductory lecture notes on the moduli space of elliptic curves over the complex numbers include

  • Richard Hain, Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)

  • section 4 of Introduction to Orbifolds (pdf)

Accounts of the general case include

  • Nicholas M. Katz, Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies_, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR MR772569 (86i:11024)

  • Joseph Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)

Lecture notes/talk notes reviewing this include

  • James Parson, Moduli of elliptic curves (pdf)

  • Akhil Mathew, section 3 of The homotopy groups of TMFTMF (pdf)

  • Andre Henriques, The moduli stack of elliptic curves (pdf) in Topological modular forms Talbot workshop 2007 (web)

For more of the general picture in view of elliptic cohomology and tmf see also

The orbifold Euler characteristic of the moduli space of curves was originally computed in

  • Don Zagier, John Harer, The Euler characteristic of the moduli space of curves, Inventiones mathematicae (1986) Volume: 85, page 457-486 (EUDML)

Reviews of the orbifold Euler characteristic computation include

  • Mathematical ideas and notions in quantum field theory – 5. The Euler characteristic of the moduli space of curves (pdf)

Revised on April 10, 2014 02:49:05 by Urs Schreiber (