We develop methods for computing the restriction map
from the cohomology of the automorphism group of a height
formal group law
(ie the height
Morava stabilizer group) to the cohomology of the automorphism group of an
–height
formal
–module, where
is the ring of
integers in a degree
field extension of .
We then compute this map for the quadratic extensions of
and the
height Morava stabilizer
group at primes
.
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
–height
–formal
–module
with the ramified part of the abelianization of the absolute Galois group of
, yielding an action of
on the Lubin–Tate/Morava
–theory spectrum
for each quadratic
extension
.
Finally, we run the associated descent spectral sequence to compute the
–homotopy
groups of the homotopy fixed-points of this action; one consequence is that, for each element in the
–local homotopy
groups of
,
either that element or an appropriate dual of it is detected in the Galois
cohomology of the abelian closure of some quadratic extension of
.
Keywords
formal groups, class field theory, stable homotopy groups,
Lubin–Tate theory, formal modules, formal groups with
complex multiplication, Morava stabilizer groups