  We study the C_p-equivariant Tate construction on the topological Hochschild
homology THH(B) of a symmetric ring spectrum B by relating it to a topological
version R_+(B) of the Singer construction, extended by a natural circle action.
This enables us to prove that the fixed and homotopy fixed point spectra of
THH(B) are p-adically equivalent for B = MU and BP. This generalizes the
classical C_p-equivariant Segal conjecture, which corresponds to the case B =
S.
