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We generalize the definition of CM cycles beyond the small and big CM onesstudied by various authors and give a uniform formula for the CM values ofGreen functions associated to these special cycles in general using the idea ofregularized 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 curvecryptography.

CM Values of Green Functions Associated to Special Cycles on Shimura Varieties with Applications to Siegel 3-Fold $X_2(2)$