---
tag: 1JM66EN5R2
kind: definition
name: Elliptic cohomology
msc: [55N34, 55N22]
defines: XH68V2BP4K
uses: [H28PSZR7XG, TNHP7XXYZT, SJ2BVG9F4W, 0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_cohomology
---

## Definition

Let $E$ be a multiplicative [[generalized cohomology theory|H28PSZR7XG]] and let $R = E^0(\mathrm{pt})$, a [[commutative ring|0310REJJY8]].
An *elliptic cohomology theory* is a triple $(E, C, \phi)$ such that:

1. $E$ is even periodic: $E^i(\mathrm{pt}) = 0$ for all odd $i$, and $E^2(\mathrm{pt})$ contains a unit of the graded ring $E^*(\mathrm{pt})$.
2. $C$ is an [[elliptic curve|SJ2BVG9F4W]] over $R$.
3. $\phi$ is an isomorphism of formal groups over $R$ from $\operatorname{Spf} E^0(\C P^\infty)$ to the formal completion $\hat C$ of $C$ at its identity.

## Remarks

In coordinates, condition 3 says the following for a suitable complex orientation of $E$.
The [[formal group law|TNHP7XXYZT]] of $E$ is the expansion of the group law of $C$ in a local parameter at the identity.

Example: let $C$ be an elliptic curve over $R$ such that the map from $\operatorname{Spec} R$ to the moduli stack of elliptic curves is flat, for instance étale.
Choose a coordinate at the identity of $C$, so that its group law gives a formal group law $F$ over $R$, and let $u$ have cohomological degree $-2$.
Then the formal group law $u^{-1} F(u x, u y)$ over $R[u, u^{-1}]$ satisfies the hypothesis of the Landweber exact functor theorem.
So $E^*(X) = MU^*(X) \otimes_{MU^*} R[u, u^{-1}]$ is a cohomology theory, it is even periodic with the unit $u^{-1}$ in degree $2$, and its formal group is $\hat C$ by construction.
An earlier, closely related theory was built by Landweber, Ravenel and Stong from the Jacobi quartic.

Non-example: ordinary cohomology $H^*(-; \Z)$ fails condition 1, since $H^2(\mathrm{pt}; \Z) = 0$ contains no unit.

Complex K-theory is even periodic, but its formal group is the multiplicative one over $\Z$, and there is no elliptic curve over $\Z$ at all.