Moduli of formal $A$–modules under change of $A$

Abstract

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (ie the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$–height $n$ formal $A$–module, where $A$ is the ring of integers in a degree $d$ field extension of ${\mathbb{Q}}_p$. We then compute this map for the quadratic extensions of ${\mathbb{Q}}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an $A$–height $1$–formal $A$–module with the ramified part of the abelianization of the absolute Galois group of $K$, yielding an action of $Gal\left(K^{ab}∕K^{nr}\right)$ on the Lubin–Tate/Morava $E$–theory spectrum $E_2$ for each quadratic extension $K∕{\mathbb{Q}}_p$. Finally, we run the associated descent spectral sequence to compute the $V\left(1\right)$–homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the $K\left(2\right)$–local homotopy groups of $V\left(1\right)$, either that element or an appropriate dual of it is detected in the Galois cohomology of the abelian closure of some quadratic extension of ${\mathbb{Q}}_p$.