Elliptic cohomology

Remarks

In coordinates, condition 3 says the following for a suitable complex orientation of $E$. The formal group law 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.