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An arithmetic intersection formula on Hilbert modular surfaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 5, October 2010
- pp. 1275-1309
- 10.1353/ajm.2010.0002
- Article
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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 Bruinier,
and is a generalization of the beautiful factorization formula of Gross
and Zagier on singular moduli. As an application, we proved the first
nontrivial non-abelian Chowla-Selberg formula, a special case of Colmez
conjecture.