  We give a new proof of a slightly weaker form of a theorem of P. Colmez. This
theorem gives a formula for the Faltings height of abelian varieties with
complex multiplication by a C.M. field whose Galois group over $\bf Q$ 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 the first paper of the series. Our proof is
intrinsic in the sense that it does not rely on the computation of the periods
of any particular abelian variety.
