nLab moduli stack of elliptic curves

Idea

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

This is formalized in algebraic geometry, hence $X$ 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 $Spec(R)$ to the groupoid of suitable elliptic curves over $Spec(R)$. 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 $\mathcal{M}_{ell}$ has a compactification $\mathcal{M}_{\overline{ell}}$ 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 $\mathcal{M}_{cub}$ of cubic curves, inside which $\mathcal{M}_{ell}$ 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), $\mathcal{M}_{ell}$ is equivalently the moduli stack of algebraic curves for genus $g = 1$ with $n = 1$ punctures, and as such is often equivalently written

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

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 $Spec(\mathbb{C}) \to \mathcal{M}_{ell}$. These are just complex tori/Riemann surfaces 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 is equivalently the moduli space of curves $\mathcal{M}_{(1,1)}$ which in turn is a quotient of the Teichmüller space $\mathcal{T}_{(1,1)}$.) This case is considered below in

• Description over the complex numbers.

Below that is the

• Description over general schemes.

as an algebraic stack $\mathcal{M}_{ell}$. 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 $\mathcal{O}$ of $\mathcal{M}_{ell}$ to a higher structure sheaf $\mathcal{O}^{top}$ of E-∞ rings in a way that makes $(\mathcal{M}_{ell}, \mathcal{O}^{op})$ a spectral Deligne-Mumford stack. The global sections of this structure sheaf yield the spectrum tmf of topological modular forms:

• As a derived scheme in E-infinity geometry

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

Description over the complex numbers

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

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

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 $\mathfrak{h}$ 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 $\mathbb{C}$

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

and we have

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

where the special linear group over the integers

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

acts as the modular group by Möbius transformations

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

The naive moduli space and its problems

Definition

Write

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

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

Definition

A holomorphic family of elliptic curves over a complex manifold $T$ is

• a holomorphic function $\pi : X \to T$

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

For every family

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

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

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

where

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

such that

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

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

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

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

For instance

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

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

consider

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

given by

$$(m,n) : (z,\tau) \mapsto (z + m \tau + n, \tau)$$

Then consider the family

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

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

and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with

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

is a family of framed elliptic curves.

Proposition

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

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

with pullback of the universal family

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

claim for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose 1-forms $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.

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

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

at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$

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

Moduli stack/orbifold of elliptic curves

Definition

Consider the global quotient stack orbifold

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

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

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

of elliptic curves

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

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

e.g. (Lurie, section 4.3).

Description over general schemes

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

$$y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6$$

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

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

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

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

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 $\mathcal{M}_{\overline{ell}}$ as a geometric stack are usefully reviewed in (Mathew, section 4).

As a derived scheme in $E_\infty$-geometry

By the Goerss-Hopkins-Miller theorem the structure sheaf $\mathcal{O}$ of the moduli stack of elliptic curves lifts to a sheaf $\mathcal{O}^{top}$ 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

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

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

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

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

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 $KO[ [q] ] \simeq KU[ [q] ]/\mathbb{Z}_2$ by forming the homotopy pushout

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

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

Properties

Finite covers

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

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

Cohomology

Proposition

$$H_1(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12\mathbb{Z}$$

$$H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0$$

$$H^2(\mathcal{M}_{1,1}, \mathbb{Z}) = \mathbb{Z}/12 \mathbb{Z}$$

$$H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q}) \simeq H_\bullet(M_{1,1}, \mathbb{Q})$$

and similarly for integral cohomology

$$\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}$$

$$Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}$$

Euler characteristic

The orbifold Euler characteristic of the moduli space of complex elliptic curves is given by the special value of the Riemann zeta function at $s= -1$

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

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.

Related concepts

• modular form

• elliptic cohomology

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

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

covering					by of level-n structures (modular curve)
	$\ast = Spec(\mathbb{Z})$	$\to$	$Spec(\mathbb{Z}[ [q] ])$	$\to$	$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering	$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack	$\mathcal{M}_{1dTori}$	$\hookrightarrow$	$\mathcal{M}_{Tate}$	$\hookrightarrow$	$\mathcal{M}_{\overline{ell}}$ (M_ell)	$\hookrightarrow$	$\mathcal{M}_{cub}$	$\to$	$\mathcal{M}_{fg}$ (M_fg)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)		(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)		tmf		$\mathbb{S}$

References

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 $TMF$ (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

• Jacob Lurie, A Survey of Elliptic Cohomology

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)