In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
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History and motivation
Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if
Definitions and constructions
Call a cohomology theory
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 law 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:
- R needs to be flat over
Z - There is no irreducible component X of
Spec R / p R , where the fiberE x x ∈ X
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.