Special Values of Green Functions at Big CM Points

Abstract

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big complex multiplication (CM) points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the Siegel–Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin–Selberg-type convolution.