  We generalize the definition of CM cycles beyond the small and big CM ones
studied by various authors and give a uniform formula for the CM values of
Green functions associated to these special cycles in general using the idea of
regularized theta lifts. Finally, as an application to Siegel 3-fold $X_2(2)$,
we can compute special values of theta functions and Rosenhain
$\lambda$-invariants at a CM cycle, which is useful for genus two curve
cryptography.
