Colmez's conjecture on Faltings heights of CM abelian varieties

Let $A$ be an abelian variety of dimension $d$ with complex multiplication by the maximal order of a CM field $E$, let $K$ be a Galois extension of ${\Bbb Q}$ containing $E$ and all embeddings of $E$ into ${\Bbb C}$, and let $G={\rm Gal}(E|{\Bbb Q})$. Let $\Phi$ be the CM type of $A$, viewed as a characteristic function on $G$, and define $\Phi^{\vee}(\tau)=\Phi(\tau^{-1})$. For complex-valued functions on $G$, write

$$\langle f,g\rangle={1\over \#G}\sum_{\gamma\in G}f(\gamma)\overline{g(\gamma)},\qquad (f\ast g)(\lambda)={1\over \#G}\sum_{\gamma\in G}f(\gamma)g(\gamma^{-1}\lambda).$$

Here $h_{\rm Fal}(A)$ is the Faltings height, and $f_{\chi}$ and $L(\chi,s)$ denote the conductor and Artin $L$-function of an irreducible Artin character $\chi$ of $G$.

Colmez's conjecture. The identity

$${1\over d}h_{\rm Fal}(A)=-\sum_{\chi\ {\rm odd}}\langle\Phi\ast\Phi^{\vee},\chi\rangle\left[2{L'(\chi,0)\over L(\chi,0)}+\log(f_{\chi})\right]$$

holds, where the sum runs over all odd irreducible Artin characters of $G$.

The conjecture is a higher-dimensional analogue of the Chowla–Selberg formula. It is known when $G$ is abelian, including without the earlier ramification restriction, but remains open when $E/{\Bbb Q}$ is nonabelian.