An arithmetic intersection formula on Hilbert modular surfaces

TL;DR

This paper derives an explicit arithmetic intersection formula on Hilbert modular surfaces, confirming a special case of a conjecture and generalizing Gross-Zagier's factorization formula, with applications to non-abelian Chowla-Selberg formulas.

Contribution

It provides a new explicit intersection formula on Hilbert modular surfaces and verifies a specific case of a conjecture related to CM cycles and modular forms.

Findings

Confirmed a special case of the author's conjecture with Bruinier.

Generalized Gross-Zagier's factorization formula for singular moduli.

Proved the first non-trivial non-abelian Chowla-Selberg formula.

Abstract

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.