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

TL;DR

This paper investigates the distribution of certain CM fields called $G$-Weyl fields, establishes their asymptotic density among all CM fields, and applies these results to confirm the Colmez conjecture for most CM fields.

Contribution

It proves asymptotic formulas for the distribution of $G$-Weyl CM fields and demonstrates the Colmez conjecture holds for almost all CM fields of a given degree.

Findings

Asymptotic density of $G$-Weyl CM fields among all CM fields is 100%

Colmez conjecture is true for a generic CM field based on distribution results

Provides explicit constants and error bounds in distribution formulas

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.