A First Look at an Equivariant Elliptic Cohomology

Usually, besides the information preserved by the formal group law of the elliptic curve, we can’t see any information about the elliptic curve when looking at the output of its associated cohomology theory.  The formal group law only* remembers if the curve was singular/supersingular, and the characteristic of its field.

(*There’s also some crazy invocation of the Serre-Tate Theorem going on s.t. operations upstairs between elliptic curve formal group laws occur downstairs between elliptic cohomology theories but that’s for another post.)

A dream is to have a cohomology theory $E^*(-)$ that encodes all of the geometry of the elliptic curve $C$ used to construct it. One way to phrase this more precisely is by asking for a cohomology theory with a property along the lines of $\textbf{E}^*(pt) = C$.

This property is inspired by an analogue to K-theory, which satisfies something like $\textbf{K}_G(pt) = \mathbb{G}_m$, where $\textbf{K}_G(-) := \text{Spec } K_G(-)$.

Turns out that I’m not alone in this desire, such a cohomology theory has been constructed before! Continue reading A First Look at an Equivariant Elliptic Cohomology