The twisted number field counting conjecture

Let $k$ be a number field, let $G$ be a transitive permutation group of degree $n$, and let $N\normal G$. Write $q:G\to G/N$ for the quotient map and

$$q_*:\operatorname{Sur}(G_k,G)\to\operatorname{Sur}(G_k,G/N)$$

for the induced pushforward. The twisted number field counting conjecture. For each $\pi\in q_*\operatorname{Sur}(G_k,G)$, there exist positive constants $a,b,c>0$, depending on $k$, $G$, $N$, and $\pi$, such that

$$\#\{\psi\in q_*^{-1}(\pi):|\disc(\psi)|\leq X\}\sim cX^{1/a}(\log X)^{b-1}$$

as $X\to\infty$. This conjecture gives the precise asymptotic count for the fibers of the map from surjections onto $G$ to surjections onto $G/N$. The paper uses it as an input in counting extensions whose minimum-index elements lie in proper abelian normal subgroups; its general validity remains open.