Toward the homotopy groups of the higher real $K$-theory $EO_2$
Vassily Gorbounov
Peter Symonds
The higher real $K$-theories $EO_n$ have been constructed by Hopkins and
Miller recently. When $n=2$ this construction suggests
a way of defining an "integral elliptic cohomology", which yields
the usual elliptic cohomology when 6 is inverted.
Our main result is the calculation of the initial term of the spectral
sequence $H^*(G,E)^{\mbox{\rm \small Gal}} \Rightarrow \pi _*EO_2$,
which converges to the homotopy groups of the "elliptic cohomology"
spectrum localized at
the prime 3. Here $G$ is the group $\Bbb Z/3\rtimes\Bbb Z/4$ and $E$ is an
infinitely generated module for $G$ over $\Bbb Z_3$, which arises from the
theory of formal groups. We also show how the integral modular forms of
Deligne appear naturally in this initial term. This calculation was
originally sketched by Hopkins and Miller, but the details
were never published, so we have proceeded by our own methods.