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Height four formal groups with quadratic complex multiplication

Andrew Salch

Algebraic & Geometric Topology 21 (2021) 2141–2173

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 p(p) 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(4)–local E4–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 E4.

formal groups, formal modules, stable homotopy, Morava stabilizer groups, stable homotopy groups
Mathematical Subject Classification 2010
Primary: 11S31, 14L05, 55N22, 55P42
Received: 14 July 2016
Revised: 31 August 2020
Accepted: 23 December 2020
Published: 31 October 2021
Andrew Salch
Department of Mathematics
Wayne State University
Detroit, MI
United States