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Tate curve

Contents

Context

Complex geometry

Arithmetic geometry

Elliptic cohomology

Contents

Idea

The Tate curve is the formal neighbourhood of the nodal curve inside the compactified moduli stack of elliptic curves ell¯\mathcal{M}_{\overline{ell}}.

Over a given base ring RR it is an elliptic curve over the power series R[[q]]R [ [ q ] ] which may be thought of as the quotient of the multiplicative group by q q^{\mathbb{Z}}.

Properties

Relation to complex K-theory

The Tate curve is naturally an oriented elliptic curve? and as such classified by a map

Spec(KU((q))) Der Spec \left( KU ( ( q ) ) \right) \longrightarrow \mathcal{M}^{Der}

in derived algebraic geometry, from the spectrum of the power series over the KU E-∞ ring to the derived moduli stack of derived elliptic curves.

Relation to Tate K-theory

The generalized cohomology theory over the Tate curve by the Goerss-Hopkins-Miller theorem is Tate K-theory KO[[q]]KO[ [q] ].

This is the homotopy limit over copies KO{1,q,,q n} +KO \wedge \{1, q, \cdots, q^n\}_+ of KO with formal parameters adjoined.

Relation to the Witten genus

The elliptic genus associated with the Tate curve is, according to the above, a formal power series of K-theory classes. This is the Witten genus (Hopkins 94).

Physically, Witten obtained this as the partition function of the heterotic string in perturbation theory about configurations where all the worldsheet sits in a single point of target space. This limiting case of the worldsheet is the Tate curve.

Formally this is given by the homotopy groups of the String orientation of tmf MStringtmfM String \to tmf followed by the map to Tate K-theory above, discussed here.

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}

References

Conceptual discussion is in

Details are spelled out in

Basic reviews include

The relation to the Witten genus originates around

  • Michael Hopkins, Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)

More details on this are in (Hill-Lawson 13, appendix A).

Last revised on November 12, 2015 at 03:12:09. See the history of this page for a list of all contributions to it.