A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of CM abelian varieties

摘要

We give a new proof of a slightly weaker form of a theorem of P. Colmez ([C2], Par. 2). This-theorem (Corollary 5.8) gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over Q is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in [KR2]. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety. [References: 34]