Memoirs of the American Mathematical Society 2009; 136 pp; softcover Volume: 204 ISBN10: 082184539X ISBN13: 9780821845394 List Price: US$74 Individual Members: US$44.40 Institutional Members: US$59.20 Order Code: MEMO/204/958
 The authors apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type \(U(1,n1)\). These cohomology theories of topological automorphic forms (\(\mathit{TAF}\)) are related to Shimura varieties in the same way that \(\mathit{TMF}\) is related to the moduli space of elliptic curves. Table of Contents  \(p\)divisible groups
 The HondaTate classification
 Tate modules and level structures
 Polarizations
 Forms and involutions
 Shimura varieties of type \(U(1,n1)\)
 Deformation theory
 Topological automorphic forms
 Relationship to automorphic forms
 Smooth \(G\)spectra
 Operation on \(\mathit{TAF}\)
 Buildings
 Hypercohomology of adele groups
 \(K(n)\)local theory
 Example: chromatic level \(1\)
 Bibliography
 Index
