A counter example to Malle’s conjecture on the asymptotics of discriminants. (English) Zbl 1083.11069

Let \(n\) be a natural number, \(G\) a transitive permutation group in \(S_n\) and \(k\) a number field. \(Z(k,G;x)\) denotes the number of extensions \(K/k\) of degree \(n\) such that the normal closure of \(K/k\) has Galois group \(G\) and the absolute norm of the discriminant \(d{K/k}\) is smaller or equal to \(x\). G. Malle [J. Number Theory 92, No. 2, 315–329 (2002; Zbl 1022.11058), Exp. Math. 13, No. 2, 129–135 (2004; Zbl 1099.11065)] has given a precise conjecture about the asymptotic behavior of the function \(Z(k,G;x)\). The author shows that the wreath product of \(C_3\) with \(C_2\) is a counter example for \(k=\mathbb Q\).


11R32 Galois theory
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[1] K. Belabas, Paramétrisation de structures algébriques et densité de discriminants [d’après Bhargava], Séminaire Bourbaki, 56ème année (935), 2004
[2] Cohen, H.; Diaz y Diaz, F.; Olivier, M., A survey of discriminant counting, (), 80-94 · Zbl 1058.11076
[3] Davenport, H.; Heilbronn, H.A., On the density of discriminants of cubic fields. II, Proc. roy. soc. London ser. A, 322, 1551, 405-420, (1971) · Zbl 0212.08101
[4] Ellenberg, J.; Venkatesh, A., The number of extensions of a number field with fixed degree and bounded discriminant, 2003
[5] Ellenberg, J.; Venkatesh, A., Counting extensions of function fields with bounded discriminant and specified Galois group, (), 151-168 · Zbl 1085.11057
[6] Klüners, J.; Malle, G., Counting nilpotent Galois extensions, J. reine angew. math., 572, 1-26, (2004) · Zbl 1052.11075
[7] Malle, G., On the distribution of Galois groups, J. numer. theory, 92, 315-322, (2002) · Zbl 1022.11058
[8] Malle, G., On the distribution of Galois groups II, Exp. math., 13, 129-135, (2004) · Zbl 1099.11065
[9] Wright, D., Distribution of discriminants of abelian extensions, Proc. London math. soc., 58, 17-50, (1989) · Zbl 0628.12006
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