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# The Chowla—Selberg Formula and The Colmez Conjecture

Published:2009-12-04
Printed: Apr 2010
• Tonghai Yang
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## Abstract

In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
 MSC Classifications: 11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22] 11F41 - Automorphic forms on ${\rm GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 14K22 - Complex multiplication [See also 11G15]