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This is an announcement of some new computational methods in stable homotopytheory, in particular, methods for using the cohomology of small-height Moravastabilizer groups to compute the cohomology of large-height Morava stabilizergroups. As an application, the cohomology of the height four Morava stabilizergroup is computed at large primes (its rank turns out to be $3440$).Consequently we are able to formulate a plausible conjecture on the rank of thelarge-primary cohomology of the Morava stabilizer groups at all heights.

The cohomology of the height four Morava stabilizer group at large primes