Height four formal groups with quadratic complex multiplication

Salch, Andrew

doi:10.2140/agt.2021.21.2141

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Height four formal groups with quadratic complex multiplication

Andrew Salch

Algebraic & Geometric Topology 21 (2021) 2141–2173

DOI: 10.2140/agt.2021.21.2141

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 $ℚ_{p} (\sqrt{p})$ 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 $E_{4}$ –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

References

Publication

Received: 14 July 2016

Revised: 31 August 2020

Accepted: 23 December 2020

Published: 31 October 2021

Authors

Andrew Salch
Department of Mathematics Wayne State University Detroit, MI United States

Volume 21, issue 5 (2021)