Special Correspondences of CM Abelian Varieties and Eisenstein Series

TL;DR

This paper establishes a connection between special divisors on stacks of CM Abelian varieties with nearby CM-types and the Fourier coefficients of derivatives of Hilbert Eisenstein series, revealing deep arithmetic and automorphic relationships.

Contribution

It introduces a new correspondence linking arithmetic divisors on CM Abelian varieties to Fourier coefficients of Eisenstein series derivatives, expanding understanding of automorphic forms and arithmetic geometry.

Findings

Arithmetic degrees of divisors relate to Fourier coefficients of Eisenstein series derivatives.

Established explicit formulas connecting CM Abelian varieties and automorphic forms.

Enhanced understanding of the arithmetic significance of Eisenstein series in CM theory.

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.