The distribution of number fields with wreath products as Galois groups
Let \(K/k\) be an extension of number fields of degree \(n\), then, by an abuse of notation, \(\mathrm{Gal}(K/k)\) is set to be the Galois group of the normal closure of \(K/k\), which is a subgroup of \(S_n\). For a given subgroup \(G\) of \(S_n\) and a number field \(k\), the author considers the number NEWLINE\[NEWLINEZ(k,G;x)=\#\{K/k:\mathrm{Gal}(K/k)=G,\mathcal N(d_{K/k})\leq x\}NEWLINE\]NEWLINE of all the extensions of \(k\) of degree \(n\) with Galois group \(G\) and such that the norm over \(\mathbb Q\) of the discriminant is bounded by \(x\). \textit{G. Malle} conjectured in [J. Number Theory 92, No. 2, 315--329 (2002; Zbl 1022.11058); Exp. Math. 13, No. 2, 129--135 (2004; Zbl 1099.11065)] that for some explicitly defined \(a(G), b(k,G)\in\mathbb R\) there exists a constant \(c(k,G)>0\) such that NEWLINE\[NEWLINEZ(k,G;x)\sim c(k,G)x^{a(G)}\log(x)^{b(k,G)-1}.NEWLINE\]NEWLINE This conjecture is known to hold for abelian groups, for \(S_3\leq S_3\) and for \(D_4\leq S_4\), but in [C. R., Math., Acad. Sci. Paris 340, No. 6, 411--414 (2005; Zbl 1083.11069)], the author showed that the conjecture does not hold for \(k=\mathbb Q\) and \(G=C_3\wr C_2\). Nevertheless the kind of problem which occurs in the counterexample seems not to be very deep, so that some refinement of the conjecture is still expected to be true.NEWLINENEWLINEIn the paper under review the author proves the original conjecture for groups of the form \(C_2\wr H\), under the assumptions that there exists at least one extension of \(k\) with Galois group \(H\) and that \(Z(k,H;x)=O_{k,H,\epsilon}(x^{1+\epsilon})\) for all \(\epsilon>0\). The techniques used in the proof are inspired by the methods described in [\textit{H. Cohen, F. Diaz y Diaz} and \textit{M. Olivier}, Compos. Math. 133, No. 1, 65--93 (2002; Zbl 1050.11104)].
scientific article; zbMATH DE number 6023790