Special Correspondences of CM Abelian Varieties and Eisenstein Series

Abstract

Let $\mathcal{CM}_{\Phi}$ be the (integral model of the) stack of principally polarized CM Abelian varieties with a CM-type $\Phi$. Considering a pair of nearby CM-types (i.e. such that they are different in exactly one embedding) $\Phi_1, \Phi_2$, we let $X = \mathcal{CM}_{\Phi_1} \times \mathcal{CM}_{\Phi_2}$ and define arithmetic divisors $\mathcal{Z}(\alpha)$ on $X$ such that the Arakelov degree of $\mathcal{Z}(\alpha)$ is (up to multiplication by an explicit constant) equal to the central value of the $\alpha^{\text{th}}$ coefficient of the Fourier expansion of the derivative of a Hilbert Eisenstein series.