Algebraicity of higher Green functions at a CM point

TL;DR

This paper explores the algebraic properties of higher Green functions at CM points on orthogonal Shimura varieties, linking them to harmonic Maass forms and answering a longstanding question of Zagier.

Contribution

It establishes the algebraicity of higher Green function values at CM points and studies harmonic Maass forms in the context of Hilbert modular forms, extending previous conjectures.

Findings

Proved algebraicity of higher Green functions at CM points

Analyzed harmonic Maass forms for Hilbert modular forms

Answered Zagier's 1986 question on Fourier coefficients

Abstract

In this paper, we investigate the algebraic nature of the value of a higher Green function on an orthogonal Shimura variety at a single CM point. This is motivated by a conjecture of Gross and Zagier in the setting of higher Green functions on the product of two modular curves. In the process, we will study analogue of harmonic Maass forms in the setting of Hilbert modular forms, and obtain results concerning the arithmetic of their holomorphic part Fourier coefficients. As a consequence, we answer a question of Zagier in his 1986 ICM proceeding.