Contents

# Contents

## Definition

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 a solution in projective space $\mathbb{P}_S^2$ of an equation of the form

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

(the Weierstrass equation) 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.

A non-singular solution to this equation is an elliptic curve (see there for more). 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 $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$

(e.g. Mathew, section 3)

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

coveringby 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)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$KU$KU[ [q] ]$elliptic spectrumcomplex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf(KO $\hookrightarrow$ KU) = KR-theoryTate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)tmf$\mathbb{S}$

## Properties

### Covers

There is an eight-fold cover of $\mathcal{M}_{cub}$ localized at $2$ (Mathew 13, section 4.2) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the $\mathbb{Z}_2$-action on KU whose homotopy fixed points are KO).

Reviews for the case that 2 and 3 are invertible include

and specifically over the complex numbers:

• Richard Hain, section 5 of Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)

Discussion of the general case in the context of the construction of tmf is in

reviewed in

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