Topological automorphic forms

Abstract

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U ( 1 , n − 1 ) U(1,n-1) . These cohomology theories of topological automorphic forms ( T A F TAF ) are related to Shimura varieties in the same way that T M F 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 ) K(n) -local homotopy types of these cohomology theories, and determine that K ( n ) K(n) -locally these spectra are given by finite products of homotopy fixed point spectra of the Morava E-theory E n E_n by finite subgroups of the Morava stabilizer group. We construct spectra Q U ( K ) Q_U(K) for compact open subgroups K K of certain adele groups, generalizing the spectra Q ( ℓ ) Q(\ell ) studied by the first author in the modular case. We show that the spectra Q U ( K ) Q_U(K) admit finite resolutions by the spectra T A F TAF , arising from the theory of buildings. We prove that the K ( n ) K(n) -localizations of the spectra Q U ( K ) Q_U(K) are finite products of homotopy fixed point spectra of E n 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 ) Q_U(K) approximate the K ( n ) K(n) -local sphere to the same degree that the spectra Q ( ℓ ) Q(\ell ) approximate the K ( 2 ) K(2) -local sphere.