Colmez conjecture and endomorphism rings

Question

It is given by the Colmez conjecture that if $A$ has a CM-type $(k, \phi)$ and $\text{End}(A) = \mathcal{O_k}$ $$\displaystyle h_{\text{fal}}(A) = \sum_{\text{irr} \hspace{1 mm}\rho: G(k^{\text{nor}}/\mathbb{Q}) \to GL_n(\mathbb{C})} C_{\rho,\phi}\left (\dfrac{L'(1,\rho)}{L(1,\rho)} +\log f\rho\right)$$ where we are taking a sum of irreducible representations $\rho$ from the Galois group of the normal closure of $k$ over $\mathbb{Q}$ to some $GL_n(\mathbb{C}).$

My question is is it necessary for $\text{End}(A) = \mathcal{O_k}$? Can you get a formula in a different way or must you have the endomorphisms of $A$ be the full ring of integers?

Note that $\log fp$ is the log of the artin conductor of $\rho$.

Answer 1

In Colmez's formulation, it is necessary that the endomorphism ring be the maximal order $\mathcal{O}_k$. It is then proved only in special cases ($k/\mathbb{Q}$ abelian), or on average over the CM types $\phi$ (by Yuan and Zhang).

There is nonetheless an extension of the formula for CM by non-maximal orders, which requires a modification in the statement and should be possible to reduce to the case of a full endomorphism ring. For elliptic curves, the precise formula was conjectured by Kaneko and proved by Nakkajima and Taguchi (A generalization of the Chowla-Selberg formula), by an algebro-geometric reduction to the maximal order case, and then by Kaneko (A generalization of the Chowla-Selberg formula and the zeta functions of quadratic order), by a direct analytic extension of the proof of the maximal order case from the Kronecker limit formula. Only the first of these approaches is likely to extend to the higher dimensional case.