Asian Journal of Mathematics

Arithmetic intersection on a Hilbert modular surface and the Faltings height

Tonghai Yang

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1383923854

Subjects
Primary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 14K22: Complex multiplication [See also 11G15]

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

Citation

Yang, Tonghai. Arithmetic intersection on a Hilbert modular surface and the Faltings height. Asian J. Math. 17 (2013), no. 2, 335--382. https://projecteuclid.org/euclid.ajm/1383923854


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