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

Abstract

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.