Singular Moduli Refined

In this paper, we give a refinement of the work of Gross and Zagier on singular moduli. Let K1 and K2 be two imaginary quadratic fields with relatively prime discriminants d1 and d2, and let F = Q( √ d1d2). Hecke constructed a Hilbert modular Eisenstein series over F of weight 1 whose functional equation forces it to vanish at s = 0. For CM elliptic curves E1 and E2 with complex multiplication by OK1 and OK2 , we define an OF -module structure and OF -quadratic form degCM on Hom(E1, E2), which is totally positive definite and satisfies TrF/Q degCM = deg. For each totally positive α ∈ F we consider the moduli stack Xα of triples (E1, E2, j) with Ei as above and j ∈ Hom(E1, E2) with degCM(j) = α. We prove that Xα has dimension 0, and that its arithmetic degree is equal to the α-th Fourier coefficient of the central derivative of Hecke’s Eisenstein series. 2000 Mathematics Subject Classification: 11G15, 11F41, 14K22