Special cycles on toroidal compactifications of orthogonal Shimura varieties

We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.

CM values of higher automorphic Green functions for orthogonal groups

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ G s ( z 1 , z 2 ) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $$z_1$$ z 1 over all CM points of a fixed discriminant $$d_1$$ d 1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ d 2 . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ GSpin ( n , 2 ) . We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.