Dihedral Artin representations and CM fields (Q6566371)

Let \(K\) be a CM-field and \(F\) its totally real subfield. An Artin representation \(\rho\) of \(F\) induced from \(K\) is of \(D_4\)-type if the fixed field \(L\) of the \(\mathrm{Ker} \, \rho\) is a Galois extension with Galois group isomorphic to the dihedral group \(D_4\) of eight elements. Such a representation is known to be induced from three quadratic extensions of \(F\) inside \(L\). The author of this paper under review divides the representations \(\rho\) of \(D_4\)-type into two classes: one is canonically induced if \(\mathrm{Gal} (L/K)\) is a cyclic subgroup of \(\mathrm{Gal} (L/F)\) and the other is a non-canonically induced if \(\mathrm{Gal} (L/K)\) is isomorphic to the Klein four-group. Let \(\delta_{K/F}^{\text{can}} \) (resp. \(\delta_{K/F}^{\text{non}} \)) be the number of isomorphism classes of canonically (resp. non-canonically) induced \(D_4\)-type Artin representations with conductor of absolute norm less than \( x\).\N\NThis paper aims to compare these two quantities, and the author proves the following interesting result:\N\[\N\delta_{K/F}^{\text{non}} (x) \sim \frac{\pi^n}{\sqrt{d_{K/F} d_F}} \delta_{K/F}^{\text{can}} (x),\N\]\Nwhere \(n=[F : \mathbb{Q}]\), \(d_{K/F}\) is the relative discriminant of \(K/F\), and \(d_F\) is the absolute discriminant of \(F\). As a consequence, a positive proportion of \(D_4\)-type Artin representations of \(F\) are non-canonically induced from \(K\).\N\NThe author derives the above result by showing an asymptotic estimate of \(\delta_{K/F}^{\text{non}} (x)\) involving the Dedekind zeta function of \(K\) and comparing it with the one obtained by \textit{H. Cohen} et al. [Compos. Math. 133, No. 1, 65--93 (2002; Zbl 1050.11104)].\N\NThe reviewer would like to point out that Artin representations induced from different quadratic subfields occur in the \(D_4\)-isoclinic case [\textit{M. Kida} and \textit{G. Koda}, Acta Arith. 191, No. 2, 115--149 (2019; Zbl 1443.11233)], and similar estimates could be considered.