The distribution of \(G\)-Weyl CM fields and the Colmez conjecture (Q6162754)

Let us write \(K^c\) for the Galois closure of the number field \(K\). When \(E\) is a CM field, with maximal totally real subfield \(F\), we have \(\mathrm{Gal}(E^c/\mathbb{Q}) \le C_2\wr \mathrm{Gal}(F^c/\mathbb{Q})\). When \(\mathrm{Gal}(F^c/\mathbb{Q})\cong G\) and \(\mathrm{Gal}(E^c/\mathbb{Q}) \cong C_2\wr G\), the authors say that \(E\) is a \(G\)-Weyl CM field. The paper is devoted to the study of the proportion of such fields among all CM fields. Two theorems are proved in this direction, both assuming a weak version of Malle's conjecture on the growth of the number of number fields with discriminant less than \(X\), as a function of \(X\). This conjecture is known to hold in quite a few cases, so the paper also contains several unconditional results. The two theorems are also about the asymptotic behaviour of the proportion, as a function of \(X\). Theorem 1.4 says that the proportion of \(G\)-Weyl CM fields among CM fields with \(\mathrm{Gal}(F^c/\mathbb{Q})\cong G\) is asymptotically 1. Theorem 1.10 gives the asymptotic proportion of \(G\)-Weyl CM fields, for some fixed \(G\), among all CM fields; the formula involves the residues of zeta functions of all totally real number fields \(F\) with \(\mathrm{Gal}(F^c/\mathbb{Q})\cong G\). Next, the authors relate the above results to the deep Colmez conjecture, itself pertaining to the Faltings height of abelian varieties with complex multiplication. Each such variety has a ``type'' whose definition involves a CM field \(E\). Theorem 1.14, whose proof is less technical than those of the previous theorems, states that when \(E\) is a \(G\)-Weyl CM field, then the Colmez conjecture holds for the abelian varieties which are related to \(E\). As a consequence, one obtains results about the proportion of fields for which the Colmez conjecture holds (and these results are consistent with the conjecture being always true).