Faltings heights of big CM cycles and derivatives of L-functions

TL;DR

This paper derives a formula for automorphic Green functions on special rational 0-cycles (big CM points) in Shimura varieties, linking them to derivatives of Eisenstein series and harmonic weak Maass forms.

Contribution

It provides an explicit formula connecting Green functions on big CM cycles to derivatives of Eisenstein series and harmonic weak Maass forms, advancing the understanding of their arithmetic properties.

Findings

Explicit formula for Green functions in terms of Fourier coefficients.

Connection between Green functions and derivatives of Eisenstein series.

Additional terms involving Rankin-Selberg convolutions for general Maass forms.

Abstract

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big 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 Mass 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.