A counter example to Malle's conjecture on the asymptotics of discriminants

Abstract

In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. To cite this article: J. Klüners, C. R. Acad. Sci. Paris, Ser. I 340 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Un contre-exemple à la conjecture de Malle sur le nombre de corps de discriminant borné. Dans cette Note, nous donnons un contre-exemple à une conjecture de Malle, qui prédit le comportement asymptotique du nombre de corps de clôture galoisienne fixée et discriminant borné. Pour citer cet article : J. Klüners, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Connected components of Hurwitz schemes and Malle's conjecture

“…As evidence in the favor of our modification of Malle's conjecture, we show that our version is not contradicted by Klüners' counterexample from [11].…”

“…In [11], Klüners shows that Z N (Q, X) ≍ X 1/2 logX contradicting with Malle's conjecture which predicts Z N (Q, X) ≍ X 1/2 . Indeed, he proves that Z N (Q, Q(ζ 3 ), X) ≍ X 1/2 logX.…”

“…We remark that there are counterexamples in [11] which are different than the one in Corollary 4.5, but in the same spirit. Using Theorem 4.3, one can show that our result gives the right asymptotic in these cases.…”

“…Proof. Given an N -extension K/F q (t), maximal constant intermediate subfield in its Galois closure K/F q (t) might be F q (t), F q 2 (t) or F q 6 (t) corresponding to the normal subgroups of N , respectively, N , We will show that Z Another interesting example in [11] is N ∼ = (C 3 ≀ C 3 ) × C 2 ⊆ S 18 where C 3 denotes the cyclic group of order 3. Klüners shows that Z N (Q, X) << X 1/4 .…”

“…Recently, Klüners gave a counterexample to this conjecture, see [11]. Indeed, take k = Q and N = ( (123) ⊕ (456) ) ⋊ (14)(25)(36) ≤ S 6 which is isomorphic to C 3 ≀ C 2 where C r is the cyclic group with r elements.…”