  Rigid meromorphic cocycles are defined in the setting of orthogonal groups of
arbitrary real signature and constructed in some instances via a $p$-adic
analogue of Borcherds' singular theta lift. The values of rigid meromorphic
cocycles at special points of an associated $p$-adic symmetric space are then
conjectured to belong to class fields of suitable global reflex fields,
suggesting an eventual framework for explicit class field theory beyond the
setting of CM fields explored in the treatise of Shimura and Taniyama.
