Connected components of Hurwitz schemes and Malle's conjecture
Let \(K\) be a number field. It is well known that, given a positive number \(X\), the number of isomorphism classes of finite extensions \(L/K\) with fixed degree and absolute value of the norm of the discriminant \(\delta(L/K):= |N_{L/Q}D(L/K)|< X\) is finite. Let \(G\leq S_n\) be a transitive group and let \(S:= \text{Stab}_G(1)\) be the stabilizer in \(G\) of \(1\in\{1,\dots, n\}\). In [Exp. Math. 13, No. 2, 129--135 (2004; Zbl 1099.11065)] \textit{G. Malle} conjectures an asymptotic formula for the number of isomorphism classes of \(G\)-extensions \(L/K\) of a fixed number field \(K\) with \(\delta(L^S/K)< X\), where \(L^S\) is the fixed field by \(S\). Recently \textit{J. Klüners} [C. R., Math., Acad. Sci. Paris 340, No. 6, 411--414 (2005; Zbl 1083.11069)] gave a counterexample to this conjecture for \(K=\mathbb{Q}\) and \(G\) a subgroup of \(S_6\). In the paper under review, the author considers the conjecture, with evident modifications, for a function field as a base field. Let \({\mathcal Z}(X)\) be the number of degree \(n\) extensions of \(K=\mathbb{F}_q(t)\) with some specified Galois group and with discriminant bounded by \(X\). The problem of computing the asymptotics for \({\mathcal Z}(X)\) can be related to a problem of counting \(\mathbb{F}_q\)-rational points on certain Hurwitz spaces. \textit{J. S. Ellenberg} and \textit{A. Venkatesh} [Prog. Math. 235, 151--168 (2005; Zbl 1085.11057)] used this idea to develop a heuristic for the asymptotic behavior of \({\mathcal Z}(X)\) the number of geometrically connected extensions, and showed that this agrees with the conjecture of Malle for function fields. The author extends Ellenberg-Venkatesh's argument [loc. cit.] to consider the more complicated case of covers of \(\mathbb{P}^1\), which may not be geometrically connected, and shows that the resulting heuristic suggests a natural modification to Malle's conjecture which avoids the Klüners counterexample to the original conjecture.
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