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cubic curve

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Definition

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

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

Equivalently 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.

A non-singular solution to this equation is an elliptic curve (see there for more). 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})

(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)
*=Spec()\ast = Spec(\mathbb{Z})\toSpec([[q]])Spec(\mathbb{Z}[ [q] ])\to ell¯[n]\mathcal{M}_{\overline{ell}}[n]
structure group of covering /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)
moduli stack 1dTori\mathcal{M}_{1dTori}\hookrightarrow Tate\mathcal{M}_{Tate}\hookrightarrow ell¯\mathcal{M}_{\overline{ell}} (M_ell)\hookrightarrow cub\mathcal{M}_{cub}\to fg\mathcal{M}_{fg} (M_fg)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value 𝒪 Σ top\mathcal{O}^{top}_{\Sigma} of structure sheaf over curve Σ\SigmaKUKU[[q]]KU[ [q] ]elliptic spectrumcomplex oriented cohomology theory
spectrum Γ(,𝒪 top)\Gamma(-, \mathcal{O}^{top}) of global sections of structure sheaf(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}

Properties

Covers

There is an eight-fold cover of cub\mathcal{M}_{cub} localized at 22 (Mathew 13, section 4.2) which is analogous to the canonical 2-fold cover of the moduli stack of formal tori (which gives the 2\mathbb{Z}_2-action on KU whose homotopy fixed points are KO).

References

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

Last revised on December 15, 2015 at 08:30:58. See the history of this page for a list of all contributions to it.