  We investigate the behavior of the return amplitude ${\cal F}(t)=
|\langle\Psi(0)|\Psi(t)\rangle|$ following a quantum quench in a conformal
field theory (CFT) on a compact spatial manifold of dimension $d-1$ and linear
size $O(L)$, from a state $|\Psi(0)\rangle$ of extensive energy with
short-range correlations. After an initial gaussian decay ${\cal F}(t)$ reaches
a plateau value related to the density of available states at the initial
energy. However for $d=3,4$ this value is attained from below after a single
oscillation. For a holographic CFT the plateau persists up to times at least
$O(\sigma^{1/(d-1)} L)$, where $\sigma\gg1$ is the dimensionless
Stefan-Boltzmann constant. On the other hand for a free field theory on
manifolds with high symmetry there are typically revivals at times
$t\sim\mbox{integer}\times L$. In particular, on a sphere $S_{d-1}$ of
circumference $2\pi L$, there is an action of the modular group on ${\cal
F}(t)$ implying structure near all rational values of $t/L$, similarly to what
happens for rational CFTs in $d=2$.
