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  "body": "\n## Definition\n\nLet $E$ be a multiplicative [[generalized cohomology theory|H28PSZR7XG]] and let $R = E^0(\\mathrm{pt})$, a [[commutative ring|0310REJJY8]].\nAn *elliptic cohomology theory* is a triple $(E, C, \\phi)$ such that:\n\n1. $E$ is even periodic: $E^i(\\mathrm{pt}) = 0$ for all odd $i$, and $E^2(\\mathrm{pt})$ contains a unit of the graded ring $E^*(\\mathrm{pt})$.\n2. $C$ is an [[elliptic curve|SJ2BVG9F4W]] over $R$.\n3. $\\phi$ 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.\n\n## Remarks\n\nIn coordinates, condition 3 says the following for a suitable complex orientation of $E$.\nThe [[formal group law|TNHP7XXYZT]] of $E$ is the expansion of the group law of $C$ in a local parameter at the identity.\n\nExample: 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.\nChoose 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 $-2$.\nThen 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.\nSo $E^*(X) = MU^*(X) \\otimes_{MU^*} R[u, u^{-1}]$ 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.\nAn earlier, closely related theory was built by Landweber, Ravenel and Stong from the Jacobi quartic.\n\nNon-example: ordinary cohomology $H^*(-; \\Z)$ fails condition 1, since $H^2(\\mathrm{pt}; \\Z) = 0$ contains no unit.\n\nComplex 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.\n",
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