The distribution of $G$-Weyl CM fields and the Colmez conjecture

Abstract

Let $G$ be a transitive subgroup of $S_d$ and $E$ be a CM field of degree $2d$ with a maximal totally real $G$-field. If the Galois group of the Galois closure of $E$ is isomorphic to the wreath product of $C_2$ and $G$, then we say that $E$ is a $G$-Weyl CM field. Let $N_{2d}^{\textrm{Weyl}}(X,G)$ count the $G$-Weyl CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$ and define \begin{align*} N_{2d}^{\textrm{Weyl}}(X):=\sum_{G \leq S_d}N_{2d}^{\textrm{Weyl}}(X,G). \end{align*} Further, let $N_{2d}^{\textrm{cm}}(X)$ count the CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$. Assuming a weak form of the upper bound in Malle's conjecture which is known to be true in many cases, we build upon an approach of Kl\"uners to prove that \begin{align*} \frac{N_{2d}^{\textrm{Weyl}}(X,G)}{N_{2d}^{\textrm{cm}}(X)} = C(d, G) + O(X^{-\alpha(d,G)}) \end{align*} and \begin{align} \frac{N_{2d}^{\textrm{Weyl}}(X)}{N_{2d}^{\textrm{cm}}(X)} = 1 + O(X^{-\beta(d)}) \qquad \qquad (0.1) \end{align} for some explicit positive constants $C(d,G), \alpha(d,G)$, and $\beta(d)$. We then apply these distribution results to study the Colmez conjecture. Using the recently proved averaged Colmez conjecture, we deduce that the Colmez conjecture is true for $G$-Weyl CM fields. Combined with (0.1), we conclude that the Colmez conjecture is true for an asymptotic density of 100% of CM fields of degree $2d$; in other words, the Colmez conjecture is true for a random CM field.