  We apply a theorem of J. Lurie to produce cohomology theories associated to
certain Shimura varieties of type U(1,n-1). These cohomology theories of
topological automorphic forms (TAF) are related to Shimura varieties in the
same way that TMF is related to the moduli space of elliptic curves. We study
the cohomology operations on these theories, and relate them to certain Hecke
algebras. We compute the K(n)-local homotopy types of these cohomology
theories, and determine that K(n)-locally these spectra are given by finite
products of homotopy fixed point spectra of the Morava E-theory E_n by finite
subgroups of the Morava stabilizer group. We construct spectra Q_U(K) for
compact open subgroups K of certain adele groups, generalizing the spectra Q(l)
studied by the first author in the modular case. We show that the spectra
Q_U(K) admit finite resolutions by the spectra TAF, arising from the theory of
buildings. We prove that the K(n)-localizations of the spectra Q_U(K) are
finite products of homotopy fixed point spectra of E_n with respect to certain
arithmetic subgroups of the Morava stabilizer groups, which N. Naumann has
shown (in certain cases) to be dense. Thus the spectra Q_U(K) approximate the
K(n)-local sphere to the same degree that the spectra Q(l) approximate the
K(2)-local sphere.
