nLab Tate curve

Idea

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

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

Properties

Relation to complex K-theory

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

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

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 $KO[ [q] ]$.

This is the homotopy limit over copies $KO \wedge \{1, q, \cdots, q^n\}_+$ 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 $M String \to tmf$ 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:

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		$\mathbb{S}$

Related concepts

• spin orientation of Tate K-theory

References

Conceptual discussion is in

• Jacob Lurie, pages 32, 33 of A Survey of Elliptic Cohomology.

Details are spelled out in

• Michael Hill, Tyler Lawson, section 3.4 of Topological modular forms with level structure (arXiv:1312.7394)

Basic reviews include

• Wikipedia, Tate curve

See also

• Antonios Broumas, Congruences between the coefficients of the Tate curve via formal groups, Proc. Amer. Math. Soc. 128 (2000), 677-681 (doi:10.1090/S0002-9939-99-05133-3)

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).