higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A moduli stack of elliptic curves is a moduli stack of elliptic curves, hence 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
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/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 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
Below that is the
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:
See also A Survey of Elliptic Cohomology - elliptic curves for more.
An elliptic curve $E \to Spec(\mathbb{C})$ over the complex numbers is determined, up to non-canonical isomorphism, by its j-invariant
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.
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 $\mathbb{C}$
and we have
where the special linear group over the integers
acts as the modular group by Möbius transformations
Write
for the plain quotient of the upper half plane by the above group action.
A homolorphic 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
we would like to have $F \to M_{1,1}$ such that there is a pullback
where
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
but see the discussion at moduli space for a discussion of the statement “it’s the automorphisms that prevent the moduli space from existing”
consider
given by
Then consider the family
is a family of elliptic curves over $\mathfrak{h}$
and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with
is a family of framed elliptic curves.
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
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
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.
Consider the global quotient stack orbifold
of the upper half plane by the action of the special linear group over the integers.
This is the moduli stack of elliptic curves.
Consider the complex analytic parameterization over the annulus
of elliptic curves
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).
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
such that $e \colon S \to X$ is the line at infinity.
Euivalently 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 $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$
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).
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
which is the moduli stack of derived elliptic curves, in that there is a natural equivalence in E-∞ rings $A$ of the form
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
(Lurie(Survey), p. 33). Again, the underlying ordinary Deligne-Mumford stack is the ordinary $\mathcal{M}_{\overline{\ell}}$.
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).
and similarly for integral cohomology
The orbifold Euler characteristic of the moduli space of complex elliptic curves is given by the special value of the Riemann zeta function at $s= -1$
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 | 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 | $\mathbb{S}$ |
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
The orbifold Euler characteristic of the moduli space of curves was originally computed in
Reviews of the orbifold Euler characteristic computation include