Elliptic Cohomology - Definitions and Constructions

Definitions and Constructions

Call a cohomology theory even periodic if for i odd and there is an invertible element . These theories possess a complex orientation, which gives a formal group law. A particularly rich source for formal group laws are elliptic curves. A cohomology theory A with is called elliptic if it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curve E over R. The usual construction of such elliptic cohomology theories uses the Landweber exact functor theorem. If the formal group laws of E is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by

Franke has identified the condition needed to fulfill Landweber exactness:

1. R needs to be flat over
2. There is no irreducible component X of, where the fiber is supersingular for every

These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from the moduli stack of elliptic curves to the moduli stack of formal groups is flat. This gives then a presheaf of cohomology theories over the site of affine schemes flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of the topological modular forms.