  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.
