Volume 21, issue 5 (2021)

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Height four formal groups with quadratic complex multiplication

Andrew Salch

Algebraic & Geometric Topology 21 (2021) 2141–2173
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 ${ℚ}_{p}\left(\sqrt{p}\right)$ for primes $p>5$. 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 ${E}_{4}$.

Keywords
formal groups, formal modules, stable homotopy, Morava stabilizer groups, stable homotopy groups
Mathematical Subject Classification 2010
Primary: 11S31, 14L05, 55N22, 55P42