An arithmetic intersection formula on Hilbert modular surfaces

In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a non-biquadratic CM quartic field. This confirms a special case of the author's conjecture with J. Bruinier in \cite{BY}, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an application, we proved the first non-trivial non-abelian Chowla-Selberg formula, a special case of Colmez conjecture.