Height four formal groups with quadratic complex multiplication

TL;DR

This paper constructs spectral sequences to compute the cohomology of automorphism groups of height four formal groups with complex multiplication, and applies this to determine the homotopy groups of related spectra in stable homotopy theory.

Contribution

It introduces new spectral sequences for automorphism groups with complex multiplication and computes their cohomology for a specific height four formal group with quadratic complex multiplication.

Findings

Cohomological dimension of the automorphism group is 8.

Total rank of the cohomology is 80.

Homotopy groups of the fixed-point spectrum are computed.

Abstract

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with complex multiplication by the ring of integers in the field $\mathbb{Q}_p(\sqrt{p})$, for primes $p>5$. This is a large subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K(4)$-local $E_4$-Adams spectral sequence to compute the homotopy groups of the homotopy fixed-point spectrum of this group's action on the Lubin-Tate/Morava spectrum $E_4$.