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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
(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:
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 | $\mathbb{S}$ |
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:
Discussion of the general case in the context of the construction of tmf is in
reviewed in
Last revised on June 4, 2020 at 13:58:42. See the history of this page for a list of all contributions to it.