## Asian Journal of Mathematics

### Arithmetic intersection on a Hilbert modular surface and the Faltings height

Tonghai Yang

#### Abstract

In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles on a Hilbert modular surface over $\mathbb{Z}$. As applications, we obtain the first ‘non-abelian’ Chowla-Selberg formula, which is a special case of Colmez’s conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in the arithmetic Siegel modular variety of genus two; Lauter’s conjecture about the denominators of CM values of Igusa invariants; and a result about bad reduction of CM genus two curves.

#### Article information

Source
Asian J. Math., Volume 17, Number 2 (2013), 335-382.

Dates
First available in Project Euclid: 8 November 2013