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# Asian Journal of Mathematics

## Volume 17 (2013)

### Number 2

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

Pages: 335 – 382

DOI: http://dx.doi.org/10.4310/AJM.2013.v17.n2.a4

#### Author

#### 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.

#### Keywords

Hilbert modular surface, Hirzebruch-Zagier divisor, arithmetic intersection, Colmez conjecture, Igusa invariants, Faltings’ height

#### 2010 Mathematics Subject Classification

11F41, 11G15, 14K22