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

TL;DR

This paper introduces new computational methods in stable homotopy theory to calculate the cohomology of large-height Morava stabilizer groups, exemplified by the height four case at large primes, and proposes a conjecture on their ranks.

Contribution

It develops methods to compute cohomology of large-height Morava stabilizer groups using small-height groups, and computes the height four case at large primes.

Findings

Cohomology rank of height four Morava stabilizer group is 3440.

New computational techniques for large-height groups.

Formulation of a conjecture on cohomology ranks at all heights.

Abstract

This is an announcement of some new computational methods in stable homotopy theory, in particular, methods for using the cohomology of small-height Morava stabilizer groups to compute the cohomology of large-height Morava stabilizer groups. As an application, the cohomology of the height four Morava stabilizer group 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 the large-primary cohomology of the Morava stabilizer groups at all heights.