---
tag: XH68V2BP4K
kind: concept
name: Elliptic cohomology
msc: [55N34, 55N22]
specializes: [H28PSZR7XG]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_cohomology
same_as:
  wikidata: Q5365789
  wikipedia: Elliptic cohomology
---

## Summary

An elliptic cohomology theory is a [[generalized cohomology theory|H28PSZR7XG]] whose [[formal group law|TNHP7XXYZT]] comes from an [[elliptic curve|427N6S6QQW]].
Ordinary cohomology corresponds to the additive group and complex K-theory to the multiplicative group; elliptic curves are the next family of one-dimensional group varieties.
Elliptic cohomology theories detect height $2$ phenomena in chromatic homotopy theory.
They are tied to elliptic genera of manifolds and to [[topological modular forms|JN73KE96YB]].

## Explanations

### L1 human

Many cohomology theories come with a rule for adding Chern classes of line bundles, given by a power series.
For ordinary cohomology the rule is plain addition, and for K-theory it is like multiplication.
Elliptic curves are curves that also carry an addition law, and near the identity this law is a power series too.
An elliptic cohomology theory is one whose rule is the addition law of an elliptic curve.
Such theories see more of the stable homotopy groups of spheres than K-theory does.