# 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!

Thanks to Eric Peterson for graciously answering all of my questions — all of them, and Minhyong Kim for carefully listening to me and clarifying my vague thoughts toward such an object. All errors in the following are mine and mine alone.

There are beasties called equivariant cohomology theories, for example, the singular beastie is defined like this: $H_G(X) := H(X\times_G EG)$

Where $$X\times_G EG := X \times EG/(x \cdot g, e) \sim (x, g \cdot e)$$, this is the Borel construction (“colimit”).

Note, $X\times_G EG$ is NOT the some shorthand for the below fiber product (“limit”):

If you’ve never encountered equivariant stuff before and want to get a feel for it beyond a beginner ranting at you, here’s an introductory lecture by Hopkins.

To the exciting part: Grojnowski constructed an equivariant $E^*(-)$ that trivially ensures that the elliptic curve is kept in the heart of the cohomology theory as its coefficient ring, $E^*(pt) = C$.

My rough understanding of Grojnowksi’s construction is:

• input:: $X :=$ space with an $S^1$-action
• look at an elliptic curve $C$
• over each point of the elliptic curve we have the stalk $H^*_{S^1}(X^c) \otimes g$, where $g$ is an element of the Lie algebra of the curve $C$
• output:: $E^*_{S^1}(X) :=$ these stalks are glued together according to the Lie algebra of the elliptic curve $C$

I’m slipping a few things under the rug when I say that $E^*_{S^1}(-)$ takes in a space $X$ with an $S^1$ action, and is stalkwise defined over each point $c$ in the elliptic curve $C$ $$(E^*_{S^1}(X))_{(e)} := H_{S^1}(X^{c})$$