Topological automorphic forms

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 QU(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.