  We define $k$-rationalized $G$-equivariant elliptic cohomology, for a field
of characteristic zero $k$ and a compact Lie group $G$, via adelic descent. We
also give adelic descriptions of rationalized $G$-equivariant singular
cohomology and K-theory. This completes a program first proposed by Ro\c{s}u.
These descriptions are then used to obtain comparison results with periodic
cyclic homology theories defined via derived algebraic geometry.
