Density of fields with a decomposition condition

Question

Let $p$ be a fixed prime number. Roughly speaking, I am interested in the following ratio

$$ \frac{ |\{\text{ all CM number fields of degree }2g}|} { |\{\text{CM fields of degree 2g, such that p splits completely in K}\}|} $$

A possible definition could be the following: let $d_{K}$ be the discriminant of K, then we can define this ratio as

$$ \lim_{d \to \infty} \frac {|\{ \text{all CM fields of degree 2g and}\ d_{K} \le d\}|} {|\{ \text{CM fields of degree 2g such that p splits completely and}\ d_{K} \le d \}|}.$$

Was it studied by anyone? I would appreciate any reference.

Answer 1

Let $K$ be a CM field with maximal real subfield $k$. Most of the time the normal closure of $k$ will have the symmetric group $S_g$ as Galois group; thus the number of fields $k$ in which $p$ splits completely has density $1/g!$ among all of them by density theorems due to Kronecker, Frobenius and/or Chebotarev. In about half of the cases, $p$ will also split in $K$, giving $1/(2 \cdot g!)$ as a rough estimate.

It is not necessarily true, however, that the density of fields with the symmetric Galois group is $1$ when the fields are ordered by discriminant. It is my impression that the corresponding problems are still open for $g > 5$, although there might be conjectural densities in the articles by Malle etc.

Thus you should be able to get a definitve answer for $g \le 4$ using known results (check the articles by Cohen et al. on the distribution of Galois groups), and perhaps $g = 5$ using recent advances e.g. by Bhargava.