CM values of automorphic Green functions on orthogonal groups over totally real fields

TL;DR

This paper generalizes previous work on CM values by computing explicit formulas for the norms of CM values of modular forms arising from regularized theta lifts on Shimura varieties over totally real fields.

Contribution

It extends the computation of CM values and height pairings to orthogonal groups over arbitrary totally real fields, providing explicit formulas for these values.

Findings

Explicit formulas for CM value norms of modular forms.

Computation of archimedian height pairings on Shimura varieties.

Generalization of Gross--Zagier and Schofer's work to broader settings.

Abstract

Generalizing work of Gross--Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to certain quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the norms of the CM values of those meromorphic modular forms arising as regularized theta lifts of holomorphic Whittaker forms.