Height four formal groups with quadratic complex multiplication

Abstract

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups equipped with additional endomorphisms given 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 additional endomorphisms of the group law by the ring of integers in the field ${\mathbb{Q}}_p\left(\sqrt{p}\right)$ for primes . This automorphism group is a large profinite 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 $K\left(4\right)$–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 $E_4$.